introduction to many areas of higher math written by former imo contestant
None
2022-11-07
<div><div>
	<p>
<a href="https://web.evanchen.cc/upload/recent-flowchart.png">
<br />(click to enlarge)</a>
</p>
<h2><a></a>Download<a href="#download">#</a></h2>
<p><strong>Download the <a href="https://venhance.github.io/napkin/Napkin.pdf">most recent draft of Napkin</a>.</strong></p>
<h2><a></a>Project Status<a href="#status">#</a></h2>
<p>The <strong>recent v1.5</strong> is a new update which revises
many of the earlier chapters and adds new content
in real analysis, measure theory, and algebraic geometry.
It is however even more visibly incomplete,
with several chapters scheduled but not yet written.
In addition, many chapters still lack problems or solutions.</p>
<p>See link above for the most recent draft,
and <a href="https://blog.evanchen.cc/napkin/">here for a listing of periodic snapshots</a>.
You can also view the
<a href="https://github.com/vEnhance/napkin/">source code on GitHub</a>;
the most recent version is
<a href="https://venhance.github.io/napkin">automatically compiled from that source</a>.</p>
<p>I would highly appreciate any corrections, suggestions, or comments.</p>
<h2><a></a>Description<a href="#description">#</a></h2>
<p>The Napkin project is a personal exposition project of mine
aimed at making higher math accessible to high school students.
The philosophy is stated in the preamble:</p>
<blockquote>
<p>I’ll be eating a quick lunch with some friends of mine who are still in high school.
They’ll ask me what I’ve been up to the last few weeks, and I’ll tell them that I’ve been learning category theory.
They’ll ask me what category theory is about.
I tell them it’s about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves.
I’ll try to give them the standard example Gp, but then I’ll realize that they don’t know what a homomorphism is.
So then I’ll start trying to explain what a homomorphism is, but then I’ll remember that they haven’t learned what a group is.
So then I’ll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they’ve already forgotten why I was talking about groups in the first place.
And then it’s 1PM, people need to go places, and I can’t help but think:</p>
<p><em>Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all.</em></p>
<p>This book is my attempt at those forty hours.</p>
</blockquote>
<p>This project has evolved to more than just forty hours.</p>
<h2><a></a>Current Table of Contents<a href="#toc">#</a></h2>
<ol>
<li>Groups</li>
<li>Metric spaces</li>
<li>Homomorphisms and quotient groups</li>
<li>Rings and ideals</li>
<li>Flavors of rings</li>
<li>Properties of metric spaces</li>
<li>Topological spaces</li>
<li>Compactness</li>
<li>Vector spaces</li>
<li>Eigen-things</li>
<li>Dual space and trace</li>
<li>Determinant</li>
<li>Inner product spaces</li>
<li>Bonus: Fourier analysis</li>
<li>Duals, adjoint, and transposes</li>
<li>Group actions overkill AIME problems</li>
<li>Find all groups</li>
<li>The PID structure theorem</li>
<li>Representations of algebras</li>
<li>Semisimple algebras</li>
<li>Characters</li>
<li>Some applications</li>
<li>Quantum states and measurements</li>
<li>Quantum circuits</li>
<li>Shor’s algorithm</li>
<li>Limits and series</li>
<li>Bonus: A hint of p-adic numbers</li>
<li>Differentiation</li>
<li>Power series and Taylor series</li>
<li>Riemann integrals</li>
<li>Holomorphic functions</li>
<li>Meromorphic functions</li>
<li>Holomorphic square roots and logarithms</li>
<li>Measure spaces</li>
<li>Constructing the Borel and Lebesgue measure</li>
<li>Lebesgue integration</li>
<li>Swapping order with Lebesgue integrals</li>
<li>Bonus: A hint of Pontryagin duality</li>
<li>Random variables (TO DO)</li>
<li>Large number laws (TO DO)</li>
<li>Stopped martingales (TO DO)</li>
<li>Multivariable calculus done correctly</li>
<li>Differential forms</li>
<li>Integrating differential forms</li>
<li>A bit of manifolds</li>
<li>Algebraic integers</li>
<li>Unique factorization (finally!)</li>
<li>Minkowski bound and class groups</li>
<li>More properties of the discriminant</li>
<li>Bonus: Let’s solve Pell’s equation!</li>
<li>Things Galois</li>
<li>Finite fields</li>
<li>Ramification theory</li>
<li>The Frobenius element</li>
<li>Bonus: A Bit on Artin Reciprocity</li>
<li>Some topological constructions</li>
<li>Fundamental groups</li>
<li>Covering projections</li>
<li>Objects and morphisms</li>
<li>Functors and natural transformations</li>
<li>Limits in categories (TO DO)</li>
<li>Abelian categories</li>
<li>Singular homology</li>
<li>The long exact sequence</li>
<li>Excision and relative homology</li>
<li>Bonus: Cellular homology</li>
<li>Singular cohomology</li>
<li>Application of cohomology</li>
<li>Affine varieties</li>
<li>Affine varieties as ringed spaces</li>
<li>Projective varieties</li>
<li>Bonus: B'ezout’s theorem</li>
<li>Morphisms of varieties</li>
<li>Sheaves and ringed spaces</li>
<li>Localization</li>
<li>Affine schemes: the Zariski topology</li>
<li>Affine schemes: the sheaf</li>
<li>Interlude: nineteen examples of affine schemes</li>
<li>Morphisms of locally ringed spaces</li>
<li>Interlude: Cauchy’s functional equation and Zorn’s lemma</li>
<li>Zermelo-Fraenkel with choice</li>
<li>Ordinals</li>
<li>Cardinals</li>
<li>Inner model theory</li>
<li>Forcing</li>
<li>Breaking the continuum hypothesis</li>
</ol>
	</div>||||I|||| Evan Chenãé³èª¼å»·ã

 web.evanchen.cc

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 The Napkin project (v1.5)


 (click to enlarge)

 Download#

 Download the most recent draft of Napkin.

 Project Status#

 The recent v1.5 is a new update which revises many of the earlier chapters and adds new content in real analysis, measure theory, and algebraic geometry. It is however even more visibly incomplete, with several chapters scheduled but not yet written. In addition, many chapters still lack problems or solutions.

 See link above for the most recent draft, and here for a listing of periodic snapshots. You can also view the source code on GitHub; the most recent version is automatically compiled from that source.

 I would highly appreciate any corrections, suggestions, or comments.

 Description#

 The Napkin project is a personal exposition project of mine aimed at making higher math accessible to high school students. The philosophy is stated in the preamble:

 I’ll be eating a quick lunch with some friends of mine who are still in high school. They’ll ask me what I’ve been up to the last few weeks, and I’ll tell them that I’ve been learning category theory. They’ll ask me what category theory is about. I tell them it’s about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves. I’ll try to give them the standard example Gp, but then I’ll realize that they don’t know what a homomorphism is. So then I’ll start trying to explain what a homomorphism is, but then I’ll remember that they haven’t learned what a group is. So then I’ll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they’ve already forgotten why I was talking about groups in the first place. And then it’s 1PM, people need to go places, and I can’t help but think:

 Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all.

 This book is my attempt at those forty hours.

 This project has evolved to more than just forty hours.

 Current Table of Contents#

 1. Groups
 2. Metric spaces
 3. Homomorphisms and quotient groups
 4. Rings and ideals
 5. Flavors of rings
 6. Properties of metric spaces
 7. Topological spaces
 8. Compactness
 9. Vector spaces
 10. Eigen-things
 11. Dual space and trace
 12. Determinant
 13. Inner product spaces
 14. Bonus: Fourier analysis
 15. Duals, adjoint, and transposes
 16. Group actions overkill AIME problems
 17. Find all groups
 18. The PID structure theorem
 19. Representations of algebras
 20. Semisimple algebras
 21. Characters
 22. Some applications
 23. Quantum states and measurements
 24. Quantum circuits
 25. Shor’s algorithm
 26. Limits and series
 27. Bonus: A hint of p-adic numbers
 28. Differentiation
 29. Power series and Taylor series
 30. Riemann integrals
 31. Holomorphic functions
 32. Meromorphic functions
 33. Holomorphic square roots and logarithms
 34. Measure spaces
 35. Constructing the Borel and Lebesgue measure
 36. Lebesgue integration
 37. Swapping order with Lebesgue integrals
 38. Bonus: A hint of Pontryagin duality
 39. Random variables (TO DO)
 40. Large number laws (TO DO)
 41. Stopped martingales (TO DO)
 42. Multivariable calculus done correctly
 43. Differential forms
 44. Integrating differential forms
 45. A bit of manifolds
 46. Algebraic integers
 47. Unique factorization (finally!)
 48. Minkowski bound and class groups
 49. More properties of the discriminant
 50. Bonus: Let’s solve Pell’s equation!
 51. Things Galois
 52. Finite fields
 53. Ramification theory
 54. The Frobenius element
 55. Bonus: A Bit on Artin Reciprocity
 56. Some topological constructions
 57. Fundamental groups
 58. Covering projections
 59. Objects and morphisms
 60. Functors and natural transformations
 61. Limits in categories (TO DO)
 62. Abelian categories
 63. Singular homology
 64. The long exact sequence
 65. Excision and relative homology
 66. Bonus: Cellular homology
 67. Singular cohomology
 68. Application of cohomology
 69. Affine varieties
 70. Affine varieties as ringed spaces
 71. Projective varieties
 72. Bonus: B'ezout’s theorem
 73. Morphisms of varieties
 74. Sheaves and ringed spaces
 75. Localization
 76. Affine schemes: the Zariski topology
 77. Affine schemes: the sheaf
 78. Interlude: nineteen examples of affine schemes
 79. Morphisms of locally ringed spaces
 80. Interlude: Cauchy’s functional equation and Zorn’s lemma
 81. Zermelo-Fraenkel with choice
 82. Ordinals
 83. Cardinals
 84. Inner model theory
 85. Forcing
 86. Breaking the continuum hypothesis

 Source repository (git) • Revision history • Suggest edit
 Updated Mon 7 Nov 2022, 21:36:42 UTC by edbd327e5fbb
Evan Chenãé³èª¼å»·ã
https://web.evanchen.cc/napkin.html
Abs  Alg
Authors:
Alg  Lin
  Ch
Grp  Classif
Cmplx  Ana
Diff  Geo
Alg  Nt
Cat  Th
Alg  Geo
Evan  Chen

Abstract

The origin of the name "Napkin" comes from the following quote of mine. I'll be eating a quick lunch with some friends of mine who are still in high school. They'll ask me what I've been up to the last few weeks, and I'll tell them that I've been learning category theory. They'll ask me what category theory is about. I tell them it's about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves. I'll try to give them the standard example Grp, but then I'll realize that they don't know what a homomorphism is. So then I'll start trying to explain what a homomorphism is, but then I'll remember that they haven't learned what a group is. So then I'll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they've already forgotten why I was talking about groups in the first place. And then it's 1PM, people need to go places, and I can't help but think: "Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all".



• Someone tells you about the hairy ball theorem in the form "you can't comb the hair on a spherical cat" then doesn't tell you anything about why it should be true, what it means to actually "comb the hair", or any of the underlying theory, leaving you with just some vague notion in your head.

• You take a class and prove every result in full detail, and at some point you stop caring about what the professor is saying.

Presumably you already know how unsatisfying the first approach is. So the second approach seems to be the default, but I really think there should be some sort of middle ground here.

Unlike university, it is not the purpose of this book to train you to solve exercises or write proofs 1 , or prepare you for research in the field. Instead I just want to show you some interesting math. The things that are presented should be memorable and worth caring about. For that reason, proofs that would be included for completeness in any ordinary textbook are often omitted here, unless there is some idea in the proof which I think is worth seeing. In particular, I place a strong emphasis over explaining why a theorem should be true rather than writing down its proof. This is a recurrent theme of this book:

Natural explanations supersede proofs.

My hope is that after reading any particular chapter in Napkin, one might get the following out of it:

• Knowing what the precise definitions are of the main characters,

• Being acquainted with the few really major examples,

• Knowing the precise statements of famous theorems, and having a sense of why they should be true.

Understanding "why" something is true can have many forms. This is sometimes accomplished with a complete rigorous proof; in other cases, it is given by the idea of the proof; in still other cases, it is just a few key examples with extensive cheerleading. Obviously this is nowhere near enough if you want to e.g. do research in a field; but if you are just a curious outsider, I hope that it's more satisfying than the elevator pitch or Wikipedia articles. Even if you do want to learn a topic with serious depth, I hope that it can be a good zoomed-out overview before you really dive in, because in many senses the choice of material is "what I wish someone had told me before I started".

More pedagogical comments and references

The preface would become too long if I talked about some of my pedagogical decisions chapter by chapter, so Appendix A contains those comments instead.

In particular, I often name specific references, and the end of that appendix has more references. So this is a good place to look if you want further reading.

Historical and personal notes

I began writing this book in December 2014, after having finished my first semester of undergraduate at Harvard. It became my main focus for about 18 months after that, as I became immersed in higher math. I essentially took only math classes (gleefully ignoring all my other graduation requirements), and merged as much of it as I could (as well as lots of other math I learned on my own time) into the Napkin.

Towards August 2016, though, I finally lost steam. The first public drafts went online then, and I decided to step back. Having burnt out slightly, I then took a break from higher math, and spent the remaining two undergraduate years2 working extensively as a coach for the American math olympiad team, and trying to spend as much time with my friends as I could before they graduated and went their own ways.

During those two years, readers sent me many kind words of gratitude, many reports of errors, and many suggestions for additions. So in November 2018, some weeks into my first semester as a math PhD student, I decided I should finish what I had started. Some months later, here is what I have.

I am indebted to countless people for this work. Here is a partial (surely incomplete) list.

• Thanks to all my teachers and professors for teaching me much of the material covered in these notes, as well as the authors of all the references I have cited here.

A special call-out to [Ga14] , [Le14] , [Sj05] , [Ga03] , [Ll15] , [Et11] , [Ko14] , [Va17] , [Pu02] , [Go18] , which were especially influential.

• Thanks also to dozens of friends and strangers who read through preview copies of my draft, and pointed out errors and gave other suggestions. Special mention to Andrej Vuković and Alexander Chua for together catching over a thousand errors.

Thanks also to Brian Gu and Tom Tseng for many corrections. (If you find mistakes or have suggestions yourself, I would love to hear them!)

• I'd also like to express my gratitude for many, many kind words I received during the development of this project. These generous comments led me to keep working on this, and were largely responsible for my decision in November 2018 to begin updating the Napkin again.

Finally, a huge thanks to the math olympiad community, from which the Napkin (and me) has its roots. All the enthusiasm, encouragement, and thank-you notes I have received over the years led me to begin writing this in the first place. I otherwise would never have the arrogance to dream a project like this was at all possible. And of course I would be nowhere near where I am today were it not for the life-changing journey I took in chasing my dreams to the IMO. Forever TWN2!

Advice for the reader §1 Prerequisites

As explained in the preface, the main prerequisite is some amount of mathematical maturity. This means I expect the reader to know how to read and write a proof, follow logical arguments, and so on.

I also assume the reader is familiar with basic terminology about sets and functions (e. g . "what is a bijection?"). If not, one should consult Appendix E.

§2 Deciding what to read

There is no need to read this book in linear order: it covers all sorts of areas in mathematics, and there are many paths you can take. In Chapter 0, I give a short overview for each part explaining what you might expect to see in that part.

For now, here is a brief chart showing how the chapters depend on each other; again see Chapter 0 for details. Dependencies are indicated by arrows; dotted lines are optional dependencies. I suggest that you simply pick a chapter you find interesting, and then find the shortest path. With that in mind, I hope the length of the entire PDF is not intimidating.

(The text in the following diagram should be clickable and links to the relevant part.)

• Each chapter features several trickier problems at the end. Some are reasonable, but others are legitimately difficult olympiad-style problems. Harder problems are marked with up to three chili peppers ( ), like this paragraph.

In addition to difficulty annotations, the problems are also marked by how important they are to the big picture.

-Normal problems, which are hopefully fun but non-central.

-Daggered problems, which are (usually interesting) results that one should know, but won't be used directly later.

-Starred problems, which are results which will be used later on in the book. 1Several hints and solutions can be found in Appendices B and C.

Image from [Go08] §4 Paper

At the risk of being blunt, Read this book with pencil and paper.

Here's why:

Image from [Or] You are not God. You cannot keep everything in your head. 2 If you've printed out a hard copy, then write in the margins. If you're trying to save paper, grab a notebook or something along with the ride. Somehow, some way, make sure you can write. Thanks. §5 On the importance of examples I am pathologically obsessed with examples. In this book, I place all examples in large boxes to draw emphasis to them, which leads to some pages of the book simply consisting of sequences of boxes one after another. I hope the reader doesn't mind. I also often highlight a "prototypical example" for some sections, and reserve the color red for such a note. The philosophy is that any time the reader sees a definition or a theorem about such an object, they should test it against the prototypical example. If the example is a good prototype, it should be immediately clear why this definition is intuitive, or why the theorem should be true, or why the theorem is interesting, et cetera.

Let me tell you a secret. Whenever I wrote a definition or a theorem in this book, I would have to recall the exact statement from my (quite poor) memory. So instead I often consider the prototypical example, and then only after that do I remember what the definition or the theorem is. Incidentally, this is also how I learned all the definitions in the first place. I hope you'll find it useful as well.

§6 Conventions and notations

This part describes some of the less familiar notations and definitions and settles for once and for all some annoying issues ("is zero a natural number?"). Most of these are "remarks for experts": if something doesn't make sense, you probably don't have to worry about it for now.

A full glossary of notation used can be found in the appendix.

§6.i Natural numbers are positive

The set N is the set of positive integers, not including 0. In the set theory chapters, we use ω = {0, 1, . . . } instead, for consistency with the rest of the book.

§6.ii Sets and equivalence relations

This is brief, intended as a reminder for experts. Consult Appendix E for full details.

An equivalence relation on a set X is a relation ∼ which is symmetric, reflexive, and transitive. An equivalence relation partitions X into several equivalence classes. We will denote this by X/∼. An element of such an equivalence class is a representative of that equivalence class.

I always use ∼ = for an "isomorphism"-style relation (formally: a relation which is an isomorphism in a reasonable category). The only time ≃ is used in the Napkin is for homotopic paths.

I generally use ⊆ and ⊊ since these are non-ambiguous, unlike ⊂. I only use ⊂ on rare occasions in which equality obviously does not hold yet pointing it out would be distracting. For example, I write Q ⊂ R since "Q ⊊ R" is distracting.

I prefer S \ T to S -T .

The power set of S (i.e., the set of subsets of S), is denoted either by 2 S or P(S).

§6.iii Functions

This is brief, intended as a reminder for experts. Consult Appendix E for full details. Let X f -→ Y be a function:

• By f pre (T ) I mean the pre-image

f pre (T ) := {x ∈ X | f (x) ∈ T } .

This is in contrast to the f -1 (T ) used in the rest of the world; I only use f -1 for an inverse function.

By abuse of notation, we may abbreviate f pre ({y}) to f pre (y). We call f pre (y) a fiber.

• By f img (S) I mean the image

f img (S) := {f (x) | x ∈ S} .

Almost everyone else in the world uses f (S) (though f [S] sees some use, and f ′′ (S) is often used in logic) but this is abuse of notation, and I prefer f img (S) for emphasis. This image notation is not standard.

• If S ⊆ X, then the restriction of f to S is denoted f ↾ S , i.e. it is the function f ↾ S : S → Y .

• Sometimes functions f : X → Y are injective or surjective; I may emphasize this sometimes by writing f : X → Y or f : X ↠ Y , respectively.

§6.iv Cycle notation for permutations

Additionally, a permutation on a finite set may be denoted in cycle notation, as described in say https://en.wikipedia.org/wiki/Permutation#Cycle_notation. For example the notation (1 2 3 4)(5 6 7) refers to the permutation with 1 → 2, 2 → 3, 3 → 4, 4 → 1, 5 → 6, 6 → 7, 7 → 5. Usage of this notation will usually be obvious from context.

§6.v Rings

All rings have a multiplicative identity 1 unless otherwise specified. We allow 0 = 1 in general rings but not in integral domains.

All rings are commutative unless otherwise specified. There is an elaborate scheme for naming rings which are not commutative, used only in the chapter on cohomology rings: On the other hand, an algebra always has 1, but it need not be commutative.

§6.vi Choice

We accept the Axiom of Choice, and use it freely. §7 Further reading

The appendix Appendix A contains a list of resources I like, and explanations of pedagogical choices that I made for each chapter. I encourage you to check it out.

In particular, this is where you should go for further reading! There are some topics that should be covered in the Napkin, but are not, due to my own ignorance or laziness. The references provided in this appendix should hopefully help partially atone for my omissions. 3 . . 44.5 Closed and exact forms . 

. 644

Excision and relative homology 645 67. 1 The long exact sequences . 

. . 748

Affine schemes: the Zariski topology 749 78.1 Some more advertising . 

. . 801 82.6 A few harder problems to think about .

. 803

Zermelo-Fraenkel with choice 805 83.1 The ultimate functional equation . 

. 805 83.3 The language of set theory .

. 806 83.4 The axioms of ZFC .

. 809 83.6 Choice and well-ordering . 

. 816 84.4 Ordinals are "tall" .

. 817 84.5 Transfinite induction and recursion .

. . 818 84.6 

. 819 84.7 The hierarchy of sets . 

. 826 85.3 

. 826 85.4 

. 827 85.5 

. 829 85.7 I made a design decision that the first part should have a little bit of both algebra and topology: so this first chapter begins by defining a group, while the second chapter begins by defining a metric space. The intention is so that newcomers get to see two different examples of "sets with additional structure" in somewhat different contexts, and to have a minimal amount of literacy as these sorts of definitions appear over and over. 1

II. Basic Abstract Algebra.

The algebraically inclined can then delve into further types of algebraic structures: some more details of groups, and then rings and fields -which will let you generalize Z, Q, R, C. So you'll learn to become familiar with all sorts of other nouns that appear in algebra, unlocking a whole host of objects that one couldn't talk about before.

We'll also come to ideals, which generalize the GCD in Z that you might know of. For example, you know in Z that any integer can be written in the form 3a + 5b for a, b ∈ Z, since gcd(3, 5) = 1. We'll see that this statement is really a statement of ideals: "(3, 5) = 1 in Z", and thus we'll understand in what situations it can be generalized, e.g. to polynomials.

III. Basic Topology.

The more analytically inclined can instead move into topology, learning more about spaces. We'll find out that "metric spaces" are actually too specific, and that it's better to work with topological spaces, which are based on the so-called open sets. You'll then get to see the buddings of some geometrical ideals, ending with the really great notion of compactness, a powerful notion that makes real analysis tick.

One example of an application of compactness to tempt you now: a continuous function f : [0, 1] → R always achieves a maximum value. ( In contrast, f : (0, 1) → R by x → 1/x does not.) We'll see the reason is that [0, 1] is compact. §0.2 Abstract algebra IV. Linear Algebra.

In high school, linear algebra is often really unsatisfying. You are given these arrays of numbers, and they're manipulated in some ways that don't really make sense. For example, the determinant is defined as this funny-looking sum with a bunch of products that seems to come out of thin air. Where does it come from? Why does det(AB) = det A det B with such a bizarre formula? Well, it turns out that you can explain all of these things! The trick is to not think of linear algebra as the study of matrices, but instead as the study of linear maps.

In earlier chapters we saw that we got great generalizations by speaking of "sets with enriched structure" and "maps between them". This time, our sets are vector spaces and our maps are linear maps. We'll find out that a matrix is actually just a way of writing down a linear map as an array of numbers, but using the "intrinsic" definitions we'll de-mystify all the strange formulas from high school and show you where they all come from.

In particular, we'll see easy proofs that column rank equals row rank, determinant is multiplicative, trace is the sum of the diagonal entries. We'll see how the dot product works, and learn all the words starting with "eigen-". We'll even have a bonus chapter for Fourier analysis showing that you can also explain all the big buzz-words by just being comfortable with vector spaces.

V. More on Groups.

Some of you might be interested in more about groups, and this chapter will give you a way to play further. It starts with an exploration of group actions, then goes into a bit on Sylow theorems, which are the tools that let us try to classify all groups.

VI. Representation Theory.

If G is a group, we can try to understand it by implementing it as a matrix, i.e. considering embeddings G → GL n (C). These are called representations of G; it turns out that they can be decomposed into irreducible ones. Astonishingly we will find that we can basically characterize all of them: the results turn out to be short and completely unexpected.

For example, we will find out that there are finitely many irreducible representations of a given finite group G; if we label them V 1 , V 2 , . . . , V r , then we will find that r is the number of conjugacy classes of G, and moreover that

|G| = (dim V 1 ) 2 + • • • + (dim V r ) 2

which comes out of nowhere!

The last chapter of this part will show you some unexpected corollaries. Here is one of them: let G be a finite group and create variables x g for each g ∈ G. A |G| × |G| matrix M is defined by setting the (g, h)th entry to be the variable x g•h . Then this determinant will turn out to factor, and the factors will correspond to the V i we described above: there will be an irreducible factor of degree dim V i appearing dim V i times. This result, called the Frobenius determinant, is said to have given birth to representation theory.

VII. Quantum Algorithms.

If you ever wondered what Shor's algorithm is, this chapter will use the built-up linear algebra to tell you! §0.3 Real and complex analysis VIII. Calculus 101.

In this part, we'll use our built-up knowledge of metric and topological spaces to give short, rigorous definitions and theorems typical of high school calculus. That is, we'll really define and prove most everything you've seen about limits, series, derivatives, and integrals.

Although this might seem intimidating, it turns out that actually, by the time we start this chapter, the hard work has already been done: the notion of limits, open sets, and compactness will make short work of what was swept under the rug in AP calculus. Most of the proofs will thus actually be quite short. We sit back and watch all the pieces slowly come together as a reward for our careful study of topology beforehand.

That said, if you are willing to suspend belief, you can actually read most of the other parts without knowing the exact details of all the calculus here, so in some sense this part is "optional".

IX. Complex Analysis.

It turns out that holomorphic functions (complex-differentiable functions) are close to the nicest things ever: they turn out to be given by a Taylor series (i.e. are basically polynomials). This means we'll be able to prove unreasonably nice results about holomorphic functions C → C, like they are determined by just a few inputs, -their contour integrals are all zero, -they can't be bounded unless they are constant, -. . . . We then introduce meromorphic functions, which are like quotients of holomorphic functions, and find that we can detect their zeros by simply drawing loops in the plane and integrating over them: the famous residue theorem appears. ( In the practice problems, you will see this even gives us a way to evaluate real integrals that can't be evaluated otherwise.) X. Measure Theory.

Measure theory is the upgraded version of integration. The Riemann integration is for a lot of purposes not really sufficient; for example, if f is the function equals 1 at rational numbers but 0 at irrational numbers, we would hope that 1 0 f (x) dx = 0, but the Riemann integral is not capable of handling this function f .

The Lebesgue integral will handle these mistakes by assigning a measure to a generic space Ω, making it into a measure space. This will let us develop a richer theory of integration where the above integral does work out to zero because the "rational numbers have measure zero". Even the development of the measure will be an achievement, because it means we've developed a rigorous, complete way of talking about what notions like area and volume mean -on any space, not just R n ! So for example the Lebesgue integral will let us integrate functions over any measure space.

XI. Probability (TO DO).

Using the tools of measure theory, we'll be able to start giving rigorous definitions of probability, too. We'll see that a random variable is actually a function from a measure space of worlds to R, giving us a rigorous way to talk about its probabilities. We can then start actually stating results like the law of large numbers and central limit theorem in ways that make them both easy to state and straightforward to prove.

XII. Differential Geometry.

Multivariable calculus is often confusing because of all the partial derivatives. But we'll find out that, armed with our good understanding of linear algebra, that we're really looking at a total derivative: at every point of a function f : R n → R we can associate a linear map Df which captures in one object the notion of partial derivatives. Set up this way, we'll get to see versions of differential forms and Stokes' theorem, and we finally will know what the notation dx really means. In the end, we'll say a little bit about manifolds in general. §0.4 Algebraic number theory XIII. Algebraic NT I: Rings of Integers.

Why is 3 + √ 5 the conjugate of 3 -√ 5? How come the norm a + b √ 5 = a 2 -5b 2 used in Pell's equations just happens to be multiplicative? Why is it we can do factoring into primes in Z[i] but not in Z[ √ -5]? All these questions and more will be answered in this part, when we learn about number fields, a generalization of Q and Z to things like Q( √ 5) and Z[ √ 5]. We'll find out that we have unique factorization into prime ideals, that there is a real multiplicative norm in play here, and so on. We'll also see that Pell's equation falls out of this theory.

XIV. Algebraic NT II: Galois and Ramification Theory.

All the big buzz-words come out now: Galois groups, the Frobenius, and friends. We'll see quadratic reciprocity is just a shadow of the behavior of the Frobenius element, and meet the Chebotarev density theorem, which generalizes greatly the Dirichlet theorem on the infinitude of primes which are a (mod n). Towards the end, we'll also state Artin reciprocity, one of the great results of class field theory, and how it generalizes quadratic reciprocity and cubic reciprocity. §0.5 Algebraic topology XV. Algebraic Topology I: Homotopy.

What's the difference between an annulus and disk? Well, one of them has a "hole" in it, but if we are just given intrinsic topological spaces it's hard to make this notion precise. The fundamental group π 1 (X) and more general homotopy group will make this precise -we'll find a way to define an abelian group π 1 (X) for every topological space X which captures the idea there is a hole in the space, by throwing lassos into the space and seeing if we can reel them in.

Amazingly, the fundamental group π 1 (X) will, under mild conditions, tell you about ways to cover X with a so-called covering projection. One picture is that one can wrap a real line R into a helix shape and then project it down into the circle S 1 . This will turn out to correspond to the fact that π 1 (S 1 ) = Z which has only one subgroup. More generally the subgroups of π 1 (X) will be in bijection with ways to cover the space X! XVI. Category Theory.

What do fields, groups, manifolds, metric spaces, measure spaces, modules, representations, rings, topological spaces, vector spaces, all have in common? Answer: they are all "objects with additional structure", with maps between them.

The notion of category will appropriately generalize all of them. We'll see that all sorts of constructions and ideas can be abstracted into the framework of a category, in which we only think about objects and arrows between them, without probing too hard into the details of what those objects are. This results in drawing many commutative diagrams.

For example, any way of taking an object in one category and getting another one (for example π 1 as above, from the category of spaces into the category of groups) will probably be a functor. We'll unify G × H, X × Y , R × S, and anything with the × symbol into the notion of a product, and then even more generally into a limit. Towards the end, we talk about abelian categories and talk about the famous snake lemma, five lemma, and so on.

XVII. Algebraic Topology II: Homology.

Using the language of category theory, we then resume our adventures in algebraic topology, in which we define the homology groups which give a different way of noticing holes in a space, in a way that is longer to define but easier to compute in practice. We'll then reverse the construction to get so-called cohomology rings instead, which give us an even finer invariant for telling spaces apart. §0.6 Algebraic geometry XVIII. Algebraic Geometry I: Classical Varieties.

We begin with a classical study of classical complex varieties: the study of intersections of polynomial equations over C. This will naturally lead us into the geometry of rings, giving ways to draw pictures of ideals, and motivating Hilbert's nullstellensatz. The Zariski topology will show its face, and then we'll play with projective varieties and quasi-projective varieties, with a bonus detour into Bézout's theorem. All this prepares us for our journey into schemes.

XIX. Algebraic Geometry II: Affine Schemes.

We now get serious and delve into Grothendieck's definition of an affine scheme: a generalization of our classical varieties that allows us to start with any ring A and construct a space Spec A on it. We'll equip it with its own Zariski topology and then a sheaf of functions on it, making it into a locally ringed space; we will find that the sheaf can be understood effectively in terms of localization on it. We'll find that the language of commutative algebra provides elegant generalizations of what's going on geometrically: prime ideals correspond to irreducible closed subsets, radical ideals correspond to closed subsets, maximal ideals correspond to closed points, and so on. We'll draw lots of pictures of spaces and examples to accompany this.

XX. Algebraic Geometry III: Schemes (TO DO).

Not yet written! Wait for v2. §0.7 Set theory XXI. Set Theory I: ZFC, Ordinals, and Cardinals.

Why is Russell's paradox such a big deal and how is it resolved? What is this Zorn's lemma that everyone keeps talking about? In this part we'll learn the answers to these questions by giving a real description of the Zermelo-Frankel axioms, and the axiom of choice, delving into the details of how math is built axiomatically at the very bottom foundations. We'll meet the ordinal numbers and cardinal numbers and learn how to do transfinite induction with them.

XXII. Set Theory II: Model Theory and Forcing.

The continuum hypothesis states that there are no cardinalities between the size of the natural numbers and the size of the real numbers. It was shown to be independent of the axioms -one cannot prove or disprove it. How could a result like that possibly be proved? Using our understanding of the ZF axioms, we'll develop a bit of model theory and then use forcing in order to show how to construct entire models of the universe in which the continuum hypothesis is true or false.

1 Groups

A group is one of the most basic structures in higher mathematics. In this chapter I will tell you only the bare minimum: what a group is, and when two groups are the same.

§1.1 Definition and examples of groups

Prototypical example for this section: The additive group of integers (Z, +) and the cyclic group Z/mZ. Just don't let yourself forget that most groups are non-commutative.

A group consists of two pieces of data: a set G, and an associative binary operation ⋆ with some properties. Before I write down the definition of a group, let me give two examples.

Example 1.1.1 (Additive integers)

The pair (Z, +) is a group: Z = {. . . , -2, -1, 0, 1, 2, . . . } is the set and the associative operation is addition. Note that • The element 0 ∈ Z is an identity: a + 0 = 0 + a = a for any a.

• Every element a ∈ Z has an additive inverse: a + (-a) = (-a) + a = 0.

We call this group Z.

Example 1.1.2 (Nonzero rationals)

Let Q × be the set of nonzero rational numbers. The pair (Q × , •) is a group: the set is Q × and the associative operation is multiplication.

Again we see the same two nice properties.

• The element 1 ∈ Q × is an identity: for any rational number, a • 1 = 1 • a = a.

• For any rational number x ∈ Q × , we have an inverse x -1 , such that

x • x -1 = x -1 • x = 1.

From this you might already have a guess what the definition of a group is. • G has an identity element, usually denoted 1 G or just 1, with the property that 1 G ⋆ g = g ⋆ 1 G = g for all g ∈ G.

• The operation is associative, meaning (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c) for any a, b, c ∈ G.

Consequently we generally don't write the parentheses.

• Each element g ∈ G has an inverse, that is, an element h ∈ G such that

g ⋆ h = h ⋆ g = 1 G .

Remark 1. 1.4 (Unimportant pedantic point) -Some authors like to add a "closure" axiom, i.e. to say explicitly that g ⋆ h ∈ G. This is implied already by the fact that ⋆ is a binary operation on G, but is worth keeping in mind for the examples below.

Remark 1. 1 .5 -It is not required that ⋆ is commutative (a ⋆ b = b ⋆ a). So we say that a group is abelian if the operation is commutative and non-abelian otherwise.

Example 1. 1.6 (Non-Examples of groups)

• The pair (Q, •) is NOT a group. (Here Q is rational numbers.) While there is an identity element, the element 0 ∈ Q does not have an inverse.

• The pair (Z, •) is also NOT a group. (Why?)

• Let Mat 2×2 (R) be the set of 2 × 2 real matrices. Then (Mat 2×2 (R), •) (where • is matrix multiplication) is NOT a group. Indeed, even though we have an identity matrix 1 0 0 1 we still run into the same issue as before: the zero matrix does not have a multiplicative inverse.

(Even if we delete the zero matrix from the set, the resulting structure is still not a group: those of you that know some linear algebra might recall that any matrix with determinant zero cannot have an inverse.)

Let's resume writing down examples. Here are some more abelian examples of groups:

Example 1. 1 

.7 (Complex unit circle)

Let S 1 denote the set of complex numbers z with absolute value one; that is

S 1 := {z ∈ C | |z| = 1} .

Then (S 1 , ×) is a group because • The complex number 1 ∈ S 1 serves as the identity, and

• Each complex number z ∈ S 1 has an inverse 1 z which is also in S 1 , since

z -1 = |z| -1 = 1.

There is one thing I ought to also check: that z 1 × z 2 is actually still in S 1 . But this follows from the fact that |z

1 z 2 | = |z 1 | |z 2 | = 1.

Example 1. 1 

.8 (Addition mod n)

Here is an example from number theory: Let n > 1 be an integer, and consider the residues (remainders) modulo n. These form a group under addition. We call this the cyclic group of order n, and denote it as Z/nZ, with elements 0, 1, . . . . The identity is 0.

Example 1. 1 

.9 (Multiplication mod p)

Let p be a prime. Consider the nonzero residues modulo p, which we denote by (Z/pZ) × . Then ((Z/pZ) × , ×) is a group. Question 1. 1.10 . Why do we need the fact that p is prime? (Digression: the notation Z/nZ and (Z/pZ) × may seem strange but will make sense when we talk about rings and ideals. Set aside your worry for now.) Here are some non-abelian examples:

Example 1. 1.11 (General linear group)

Let n be a positive integer. Then GL n (R) is defined as the set of n × n real matrices which have nonzero determinant. It turns out that with this condition, every matrix does indeed have an inverse, so (GL n (R), ×) is a group, called the general linear group.

(The fact that GL n (R) is closed under × follows from the linear algebra fact that det(AB) = det A det B, proved in later chapters.) Example 1. 1 .12 (Special linear group) Following the example above, let SL n (R) denote the set of n × n matrices whose determinant is actually 1. Again, for linear algebra reasons it turns out that (SL n (R), ×) is also a group, called the special linear group.

Example 1. 1 .13 (Symmetric groups) Let S n be the set of permutations of {1, . . . , n}. By viewing these permutations as functions from {1, . . . , n} to itself, we can consider compositions of permutations. Then the pair (S n , •) (here • is function composition) is also a group, because • There is an identity permutation, and

• Each permutation has an inverse.

The group S n is called the symmetric group on n elements. The dihedral group of order 2n, denoted D 2n , is the group of symmetries of a regular n-gon A 1 A 2 . . . A n , which includes rotations and reflections. It consists of the 2n elements 1, r, r 2 , . . . , r n-1 , s, sr, sr 2 , . . . , sr n- 1 .

The element r corresponds to rotating the n-gon by 2π n , while s corresponds to reflecting it across the line OA 1 (here O is the center of the polygon). So rs mean "reflect then rotate" (like with function composition, we read from right to left).

In particular, r n = s 2 = 1. You can also see that r k s = sr -k .

Here is a picture of some elements of D 10 . Let (G, ⋆) and (H, * ) be groups. We can define a product group (G × H, •), as follows. The elements of the group will be ordered pairs (g, h) ∈ G × H. Then

(g 1 , h 1 ) • (g 2 , h 2 ) = (g 1 ⋆ g 2 , h 1 * h 2 ) ∈ G × H

is the group operation.

Question 1. 1.16 . What are the identity and inverses of the product group?

Example 1. 1 .17 (Trivial group)

The trivial group, often denoted 0 or 1, is the group with only an identity element. I will use the notation {1}.

Exercise 1. 1.18 . Which of these are groups?

(a) Rational numbers with odd denominators (in simplest form), where the operation is addition. (This includes integers, written as n/1, and 0 = 0/1).

(b) The set of rational numbers with denominator at most 2, where the operation is addition.

(c) The set of rational numbers with denominator at most 2, where the operation is multiplication.

(d) The set of nonnegative integers, where the operation is addition.

§1.2 Properties of groups

Prototypical example for this section: (Z/pZ) × is possibly best.

Abuse of Notation 1.2. 1 . From now on, we'll often refer to a group (G, ⋆) by just G. Moreover, we'll abbreviate a ⋆ b to just ab. Also, because the operation ⋆ is associative, we will omit unnecessary parentheses: (ab)c = a(bc) = abc.

Abuse of Notation 1.2.2. From now on, for any g ∈ G and n ∈ N we abbreviate

g n = g ⋆ • • • ⋆ g n times

.

Moreover, we let g -1 denote the inverse of g, and g -n = (g -1 ) n .

In mathematics, a common theme is to require that objects satisfy certain minimalistic properties, with certain examples in mind, but then ignore the examples on paper and try to deduce as much as you can just from the properties alone. (Math olympiad veterans are likely familiar with "functional equations" in which knowing a single property about a function is enough to determine the entire function.) Let's try to do this here, and see what we can conclude just from knowing Definition 1. 1.3 .

It is a law in Guam and 37 other states that I now state the following proposition.

Fact 1.2.3. Let G be a group.

(a) The identity of a group is unique.

(b) The inverse of any element is unique.

(c) For any g ∈ G, (g -1 ) -1 = g.

Proof. This is mostly just some formal manipulations, and you needn't feel bad skipping it on a first read.

(a) If 1 and 1 ′ are identities, then 1 = 1 ⋆ 1 ′ = 1 ′ .

(b) If h and h ′ are inverses to g, then 1

G = g ⋆ h =⇒ h ′ = (h ′ ⋆ g) ⋆ h = 1 G ⋆ h = h.

(c) Trivial; omitted.

Now we state a slightly more useful proposition.

Proposition 1.2.4 (Inverse of products)

Let G be a group, and a, b ∈ G. Then (ab) -1 = b -1 a -1 .

Proof. Direct computation. We have

(ab)(b -1 a -1 ) = a(bb -1 )a -1 = aa -1 = 1 G .

Similarly, (b -1 a -1 )(ab) = 1 G as well. Hence (ab) -1 = b -1 a -1 .

Finally, we state a very important lemma about groups, which highlights why having an inverse is so valuable.

Lemma 1.2.5 (Left multiplication is a bijection)

Let G be a group, and pick a g ∈ G. Then the map G → G given by x → gx is a bijection. Let G = (Z/7Z) × (as in Example 1. 1.9 ) and pick g = 3. The above lemma states that the map x → 3 • x is a bijection, and we can see this explicitly: The fact that the map is injective is often called the cancellation law. (Why do you think so?) Abuse of Notation 1.2.8 (Later on, sometimes the identity is denoted 0 instead of 1). You don't need to worry about this for a few chapters, but I'll bring it up now anyways. In most of our examples up until now the operation ⋆ was thought of like multiplication of some sort, which is why 1 = 1 G was a natural notation for the identity element.

But there are groups like Z = (Z, +) where the operation ⋆ is thought of as addition, in which case the notation 0 = 0 G might make more sense instead. ( In general, whenever an operation is denoted +, the operation is almost certainly commutative.) We will eventually start doing so too when we discuss rings and linear algebra. §1.3 Isomorphisms Prototypical example for this section: Z ∼ = 10Z.

First, let me talk about what it means for groups to be isomorphic. Consider the two groups • Z = ({. . . , -2, -1, 0, 1, 2, . . . } , +).

• 10Z = ({. . . , -20, -10, 0, 10, 20, . . . } , +). These groups are "different", but only superficially so -you might even say they only differ in the names of the elements. Think about what this might mean formally for a moment.

Specifically the map ϕ : Z → 10Z by x → 10x is a bijection of the underlying sets which respects the group operation. In symbols, ϕ(x + y) = ϕ(x) + ϕ(y).

In other words, ϕ is a way of re-assigning names of the elements without changing the structure of the group. That's all just formalism for capturing the obvious fact that (Z, +) and (10Z, +) are the same thing. Now, let's do the general definition.

Definition 1.3. 1 . Let G = (G, ⋆) and H = (H, * ) be groups. A bijection ϕ : G → H is called an isomorphism if ϕ(g 1 ⋆ g 2 ) = ϕ(g 1 ) * ϕ(g 2 ) for all g 1 , g 2 ∈ G.

If there exists an isomorphism from G to H, then we say G and H are isomorphic and write G ∼ = H.

Note that in this definition, the left-hand side ϕ(g 1 ⋆ g 2 ) uses the operation of G while the right-hand side ϕ(g 1 ) * ϕ(g 2 ) uses the operation of H.

Example 1.3.2 (Examples of isomorphisms)

Let G and H be groups. We have the following isomorphisms.

(a) Z ∼ = 10Z, as above.

(b) There is an isomorphism

G × H ∼ = H × G

by the map (g, h) → (h, g).

(c) The identity map id : G → G is an isomorphism, hence G ∼ = G.

(d) There is another isomorphism of Z to itself: send every x to -x.

Example 1.3.3 (Primitive roots modulo 7)

As a nontrivial example, we claim that Z/6Z ∼ = (Z/7Z) × . The bijection is ϕ(a mod 6) = 3 a mod 7.

• This map is a bijection by explicit calculation:

(3 0 , 3 1 , 3 2 , 3 3 , 3 4 , 3 5 ) ≡ (1, 3, 2, 6, 4, 5) (mod 7).

(Technically, I should more properly write 3 0 mod 6 = 1 and so on to be pedantic.)

• Finally, we need to verify that this map respects the group action. In other words, we want to see that ϕ(a + b) = ϕ(a)ϕ(b) since the operation of Z/6Z is addition while the operation of (Z/7Z) × is multiplication. That's just saying that 3 a+b mod 6 ≡ 3 a mod 6 3 b mod 6 (mod 7), which is true.

Example 1.3.4 (Primitive roots)

More generally, for any prime p, there exists an element g ∈ (Z/pZ) × called a primitive root modulo p such that 1, g, g 2 , . . . , g p-2 are all different modulo p. One can show by copying the above proof that Z/(p -1)Z ∼ = (Z/pZ) × for all primes p.

The example above was the special case p = 7 and g = 3.

Exercise 1.3.5. Assuming the existence of primitive roots, establish the isomorphism Z/(p -1)Z ∼ = (Z/pZ) × as above.

It's not hard to see that ∼ = is an equivalence relation (why?). Moreover, because we really only care about the structure of groups, we'll usually consider two groups to be the same when they are isomorphic. So phrases such as "find all groups" really mean "find all groups up to isomorphism". (c) Consider the group G × H (Example 1. 1.15 ) and the elements {(g,

1 H ) | g ∈ G}.

This is a subgroup of G × H (why?). In fact, it is isomorphic to G by the isomorphism (g, 1 H ) → g.

Example 1.5.4 (Stupid examples of subgroups)

For any group G, the trivial group {1 G } and the entire group G are subgroups of G.

Next is an especially important example that we'll talk about more in later chapters.

Example 1.5.5 (Subgroup generated by an element) Let x be an element of a group G. Consider the set ⟨x⟩ = . . . , x -2 , x -1 , 1, x, x 2 , . . . . This is also a subgroup of G, called the subgroup generated by x. , -8 , -1, 0, 1, 8, . . . } is not a subgroup because it is not closed under addition; the sum of two cubes is not in general a cube.

(c) The empty set ∅ is not a subgroup of Z because it lacks an identity element. §1.6 Groups of small orders Just for fun, here is a list of all groups of order less than or equal to ten (up to isomorphism, of course). 1 . The only group of order 1 is the trivial group.

2. The only group of order 2 is Z/2Z.

3.

The only group of order 3 is Z/3Z.

4. The only groups of order 4 are

• Z/4Z, the cyclic group on four elements,

• Z/2Z × Z/2Z, called the Klein Four Group.

5. The only group of order 5 is Z/5Z.

6. The groups of order six are

• Z/6Z, the cyclic group on six elements.

• S 3 , the permutation group of three elements. This is the first non-abelian group.

Some of you might wonder where Z/2Z × Z/3Z is. All I have to say is: Chinese remainder theorem! You might wonder where D 6 is in this list. It's actually isomorphic to S 3 .

7. The only group of order 7 is Z/7Z.

8. The groups of order eight are more numerous.

• Z/8Z, the cyclic group on eight elements.

• Z/4Z × Z/2Z.

• Z/2Z × Z/2Z × Z/2Z.

• D 8 , the dihedral group with eight elements, which is not abelian.

• A non-abelian group Q 8 , called the quaternion group. It consists of eight elements ±1, ±i, ±j, ±k with i 2 = j 2 = k 2 = ijk = -1.

9. The groups of order nine are

• Z/9Z, the cyclic group on nine elements.

• Z/3Z × Z/3Z.

10. The groups of order 10 are

• Z/10Z ∼ = Z/5Z × Z/2Z (again Chinese remainder theorem).

• D 10 , the dihedral group with 10 elements. This group is non-abelian.

§1.7 Unimportant long digression

A common question is: why these axioms? For example, why associative but not commutative? This answer will likely not make sense until later, but here are some comments that may help.

One general heuristic is: Whenever you define a new type of general object, there's always a balancing act going on. On the one hand, you want to include enough constraints that your objects are "nice". On the other hand, if you include too many constraints, then your definition applies to too few objects.

So, for example, we include "associative" because that makes our lives easier and most operations we run into are associative. In particular, associativity is required for the inverse of an element to necessarily be unique. However we don't include "commutative", because examples below show that there are lots of non-abelian groups we care about. (But we introduce another name "abelian" because we still want to keep track of it.)

Another comment: a good motivation for the inverse axioms is that you get a large amount of symmetry. The set of positive integers with addition is not a group, for example, because you can't subtract 6 from 3: some elements are "larger" than others. By requiring an inverse element to exist, you get rid of this issue. (You also need identity for this; it's hard to define inverses without it.)

Even more abstruse comment: Problem 1F † shows that groups are actually shadows of the so-called symmetric groups (defined later, also called permutation groups). This makes rigorous the notion that "groups are very symmetric". §1.8 A few harder problems to think about Problem 1A. What is the joke in the following figure? (Source: [Ge] .) Problem 1B. Prove Lagrange's theorem for orders in the special case that G is a finite abelian group.

Problem 1C. Show that D 6 ∼ = S 3 but D 24 ̸ ∼ = S 4 . Problem 1D ⋆ . Let p be a prime. Show that the only group of order p is Z/pZ. Problem 1E (A hint for Cayley's theorem). Find a subgroup H of S 8 which is isomorphic to D 8 , and write the isomorphism explicitly. Problem 1F † . Let G be a finite group. 1 Show that there exists a positive integer n such that (a) (Cayley's theorem) G is isomorphic to some subgroup of the symmetric group S n .

(b) (Representation Theory) G is isomorphic to some subgroup of the general linear group GL n (R). (This is the group of invertible n × n matrices.)

Problem 1G (IMO SL 2005 C5

). There are n markers, each with one side white and the other side black. In the beginning, these n markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it.

Prove that if n ≡ 1 (mod 3) it's impossible to reach a state with only two markers remaining. ( In fact the converse is true as well.) Problem 1H. Let p be a prime and F 1 = F 2 = 1, F n+2 = F n+1 + F n be the Fibonacci sequence. Show that F 2p(p 2 -1) is divisible by p.

At the time of writing, I'm convinced that metric topology is the morally correct way to motivate point-set topology as well as to generalize normal calculus. 1 So here is my best attempt.

The concept of a metric space is very "concrete", and lends itself easily to visualization. Hence throughout this chapter you should draw lots of pictures as you learn about new objects, like convergent sequences, open sets, closed sets, and so on.

§2.1 Definition and examples of metric spaces

Prototypical example for this section: R 2 , with the Euclidean metric. (b) The interval [0, 1] is also a metric space with the same distance function.

(c) In fact, any subset S of R can be made into a metric space in this way.

Example 2. 1.4 (Metric spaces of R 2 ) (a) We can make R 2 into a metric space by imposing the Euclidean distance function d ((x 1 , y 1 ), (x 2 , y 2 )) = (x 1 -x 2 ) 2 + (y 1 -y 2 ) 2 .

(b) Just like with the first example, any subset of R 2 also becomes a metric space after we inherit it. The unit disk, unit circle, and the unit square [0, 1] 2 are special cases.

Example 2. 1 .5 (Taxicab on R 2 )

It is also possible to place the taxicab distance on R 2 :

d ((x 1 , y 1 ), (x 2 , y 2 )) = |x 1 -x 2 | + |y 1 -y 2 | .

For now, we will use the more natural Euclidean metric.

Example 2. 1.6 (Metric spaces of R n )

We can generalize the above examples easily. Let n be a positive integer.

(a) We let R n be the metric space whose points are points in n-dimensional Euclidean space, and whose metric is the Euclidean metric

d ((a 1 , . . . , a n ) , (b 1 , . . . , b n )) = (a 1 -b 1 ) 2 + • • • + (a n -b n ) 2 .

This is the n-dimensional Euclidean space.

(b)

The open unit ball B n is the subset of R n consisting of those points (x 1 , . . . , x n ) such that

x 2 1 + • • • + x 2 n < 1.

(c) The unit sphere S n-1 is the subset of R n consisting of those points (x 1 , . . . , x n ) such that x 2 1 + • • • + x 2 n = 1, with the inherited metric. (The superscript n -1 indicates that S n-1 is an n -1 dimensional space, even though it lives in ndimensional space.) For example, S 1 ⊆ R 2 is the unit circle, whose distance between two points is the length of the chord joining them. You can also think of it as the "boundary" of the unit ball B n .

Example 2. 1 

.7 (Function space)

We can let M be the space of continuous functions f : [0, 1] → R and define the metric by d(f, g) = 1 0 |f -g| dx. (It admittedly takes some work to check d(f, g) = 0 implies f = g, but we won't worry about that yet.)

Here is a slightly more pathological example.

Example 2. 1 

.8 (Discrete space)

Let S be any set of points (either finite or infinite). We can make S into a discrete space by declaring

d(x, y) = 1 if x ̸ = y 0 if x = y.

If |S| = 4 you might think of this space as the vertices of a regular tetrahedron, living in R 3 . But for larger S it's not so easy to visualize. . .

Example 2. 1 .9 (Graphs are metric spaces)

Any connected simple graph G can be made into a metric space by defining the distance between two vertices to be the graph-theoretic distance between them. (The discrete metric is the special case when G is the complete graph on S.)

Question 2. 1.10 . Check the conditions of a metric space for the metrics on the discrete space and for the connected graph.

Abuse of Notation 2. 1.11 . From now on, we will refer to R n with the Euclidean metric by just R n . Moreover, if we wish to take the metric space for a subset S ⊆ R n with the inherited metric, we will just write S.

§2.2 Convergence in metric spaces

Prototypical example for this section: The sequence 1 n (for n = 1, 2, . . . ) in R. Since we can talk about the distance between two points, we can talk about what it means for a sequence of points to converge. This is the same as the typical epsilon-delta definition, with absolute values replaced by the distance function.

Definition 2.2. 1 . Let (x n ) n≥1 be a sequence of points in a metric space M . We say that x n converges to x if the following condition holds: for all ε > 0, there is an integer N (depending on ε) such that d(x n , x) < ε for each n ≥ N . This is written

x n → x

or more verbosely as lim n→∞

x n = x.

We say that a sequence converges in M if it converges to a point in M .

You should check that this definition coincides with your intuitive notion of "converges".

Abuse of Notation 2.2.2. If the parent space M is understood, we will allow ourselves to abbreviate "converges in M " to just "converges". However, keep in mind that convergence is defined relative to the parent space; the "limit" of the space must actually be a point in M for a sequence to converge.

x 1

x 2

x 3

x 4

x 5 x 6

x 7

x 8

x 9 x

Example 2.2.3

Consider the sequence x 1 = 1, x 2 = 1.4, x 3 = 1.41, x 4 = 1.414, . . . .

(a) If we view this as a sequence in R, it converges to √ 2.

(b) However, even though each x i is in Q, this sequence does NOT converge when we view it as a sequence in Q! Moreover, the entire function f is continuous if it is continuous at every point p ∈ M .

Notice that, just like in our definition of an isomorphism of a group, we use the metric of M for one condition and the metric of N for the other condition.

This generalization is nice because it tells us immediately how we could carry over continuity arguments in R to more general spaces like C. Nonetheless, this definition is kind of cumbersome to work with, because it makes extensive use of the real numbers (epsilons and deltas). Here is an equivalent condition.

Theorem 2.3.3 (Sequential continuity)

A function f : M → N of metric spaces is continuous at a point p ∈ M if and only if the following property holds: if x 1 , x 2 , . . . is a sequence in M converging to p, then the sequence f (x 1 ), f (x 2 ), . . . in N converges to f (p).

Proof. One direction is not too hard:

Exercise 2.3.4. Show that ε-δ continuity implies sequential continuity at each point.

Conversely, we will prove if f is not ε-δ continuous at p then it does not preserve convergence.

If f is not continuous at p, then there is a "bad" ε > 0, which we now consider fixed. So for each choice of δ = 1/n, there should be some point x n which is within δ of p, but which is mapped more than ε away from f (p). But then the sequence x n converges to p, and f (x n ) is always at least ε away from f (p), contradiction.

Example application showcasing the niceness of sequential continuity: Proposition 2. 3 

.5 (Composition of continuous functions is continuous)

Let f : M → N and g : N → L be continuous maps of metric spaces. Then their composition g • f is continuous.

Proof. Dead simple with sequences: Let p ∈ M be arbitrary and let x n → p in M . Then f (x n ) → f (p) in N and g(f (x n )) → g(f (p)) in L, QED.

Question 2.3.6. Let M be any metric space and D a discrete space. When is a map f : D → M continuous? §2.4 Homeomorphisms Prototypical example for this section: The unit circle S 1 is homeomorphic to the boundary of the unit square.

When do we consider two groups to be the same? Answer: if there's a structurepreserving map between them which is also a bijection. For metric spaces, we do exactly the same thing, but replace "structure-preserving" with "continuous".

Definition 2.4. 1. Let M and N be metric spaces. A function f : M → N is a homeomorphism if it is a bijection, and both f : M → N and its inverse f -1 : N → M are continuous. We say M and N are homeomorphic.

Needless to say, homeomorphism is an equivalence relation. You might be surprised that we require f -1 to also be continuous. Here's the reason: you can show that if ϕ is an isomorphism of groups, then ϕ -1 also preserves the group operation, hence ϕ -1 is itself an isomorphism. The same is not true for continuous bijections, which is why we need the new condition.

Example 2. 4 .2 (Homeomorphism ̸ = continuous bijection) (a) There is a continuous bijection from [0, 1) to the circle, but it has no continuous inverse.

(b) Let M be a discrete space with size |R|. Then there is a continuous function M → R which certainly has no continuous inverse.

Note that this is the topologist's definition of "same" -homeomorphisms are "continuous deformations". Here are some examples. It may have seemed strange that our metric function on S 1 was the one inherited from R 2 , meaning the distance between two points on the circle was defined to be the length of the chord. Wouldn't it have made more sense to use the circumference of the smaller arc joining the two points?

In fact, it doesn't matter: if we consider S 1 with the "chord" metric and the "arc" metric, we get two homeomorphic spaces, as the map between them is continuous.

The same goes for S n-1 for general n.

Example 2.4.5 (Homeomorphisms really don't preserve size)

Surprisingly, the open interval (-1, 1) is homeomorphic to the real line R! One bijection is given by x → tan(xπ/2) with the inverse being given by t → 2 π arctan(t). This might come as a surprise, since (-1, 1) doesn't look that much like R; the former is "bounded" while the latter is "unbounded". §2.5 Extended example/definition: product metric Prototypical example for this section: R × R is R 2 .

Here is an extended example which will occur later on. Let M = (M, d M ) and N = (N, d N ) be metric spaces (say, M = N = R). Our goal is to define a metric space on M × N .

Let p i = (x i , y i ) ∈ M × N for i = 1, 2. Consider the following metrics on the set of points M × N :

d max (p 1 , p 2 ) := max {d M (x 1 , x 2 ), d N (y 1 , y 2 )} d Euclid (p 1 , p 2 ) := d M (x 1 , x 2 ) 2 + d N (y 1 , y 2 ) 2 d taxicab (p 1 , p 2 ) := d M (x 1 , x 2 ) + d N (y 1 , y 2 ).

All of these are good candidates. We are about to see it doesn't matter which one we use:

Exercise 2.5. 1 . Verify that

d max (p 1 , p 2 ) ≤ d Euclid (p 1 , p 2 ) ≤ d taxicab (p 1 , p 2 ) ≤ 2d max (p 1 , p 2 ).

Use this to show that the metric spaces we obtain by imposing any of the three metrics are homeomorphic, with the homeomorphism being just the identity map.

Definition 2.5.2. Hence we will usually simply refer to the metric on M × N , called the product metric. It will not be important which of the three metrics we select. If M = N = R, we get R 2 , the Euclidean plane. The metric d Euclid is the one we started with, but using either of the other two metric works fine as well.

The product metric plays well with convergence of sequences.

Proposition 2.5.4 (Convergence in the product metric is by component)

We have (x n , y n ) → (x, y) if and only if x n → x and y n → y.

Proof. We have d max ((x, y), (x n , y n )) = max {d M (x, x n ), d N (y, y n )} and the latter approaches zero as n → ∞ if and only if d M (x, x n ) → 0 and d N (y, y n ) → 0.

Let's see an application of this: Proposition 2.5.5 (Addition and multiplication are continuous)

The addition and multiplication maps are continuous maps R × R → R.

Proof. For multiplication: for any n we have

x n y n = (x + (x n -x)) (y + (y n -y)) = xy + y(x n -x) + x(y n -y) + (x n -x)(y n -y) =⇒ |x n y n -xy| ≤ |y| |x n -x| + |x| |y n -y| + |x n -x| |y n -y| .

As n → ∞, all three terms on the right-hand side tend to zero. The proof that + : R × R → R is continuous is similar (and easier): one notes for any n that Problem 2C covers the other two operations, subtraction and division. The upshot of this is that, since compositions are also continuous, most of your naturally arising real-valued functions will automatically be continuous as well. For example, the function 3x

x 2 +1 will be a continuous function from R → R, since it can be obtained by composing +, ×, ÷. §2.6 Open sets Prototypical example for this section: The open disk x 2 + y 2 < r 2 in R 2 .

Continuity is really about what happens "locally": how a function behaves "close to a certain point p". One way to capture this notion of "closeness" is to use metrics as we've done above. In this way we can define an r-neighborhood of a point. Definition 2.6. 1. Let M be a metric space. For each real number r > 0 and point p ∈ M , we define M r (p) := {x ∈ M : d(x, p) < r} .

The set M r (p) is called an r-neighborhood of p. We can rephrase convergence more succinctly in terms of r-neighborhoods. Specifically, a sequence (x n ) converges to x if for every r-neighborhood of x, all terms of x n eventually stay within that r-neighborhood.

Let's try to do the same with functions.

Question 2.6.2. In terms of r-neighborhoods, what does it mean for a function f : M → N to be continuous at a point p ∈ M ?

Essentially, we require that the pre-image of every ε-neighborhood has the property that some δ-neighborhood exists inside it. This motivates: Definition 2.6.3. A set U ⊆ M is open in M if for each p ∈ U , some r-neighborhood of p is contained inside U . In other words, there exists r > 0 such that M r (p) ⊆ U .

Abuse of Notation 2.6.4. Note that a set being open is defined relative to the parent space M . However, if M is understood we can abbreviate "open in M " to just "open". Now assume f is continuous. First, suppose V is an open subset of the metric space N ; let U = f pre (V ). Pick x ∈ U , so y = f (x) ∈ V ; we want an open neighborhood of x inside U .

N y ε V f M x δ U = f pre (V )

As V is open, there is some small ε-neighborhood around y which is contained inside V . By continuity of f , we can find a δ such that the δ-neighborhood of x gets mapped by f into the ε-neighborhood in N , which in particular lives inside V . Thus the δ-neighborhood lives in U , as desired. §2.7 Closed sets Prototypical example for this section: The closed unit disk x 2 + y 2 ≤ r 2 in R 2 .

It would be criminal for me to talk about open sets without talking about closed sets. The name "closed" comes from the definition in a metric space.

Definition 2.7. 1. Let M be a metric space. A subset S ⊆ M is closed in M if the following property holds: let x 1 , x 2 , . . . be a sequence of points in S and suppose that x n converges to x in M . Then x ∈ S as well.

Abuse of Notation 2.7.2. Same caveat: we abbreviate "closed in M " to just "closed" if the parent space M is understood.

Here's another way to phrase it. The limit points of a subset S ⊆ M are defined by lim S := {p ∈ M : ∃(x n ) ∈ S such that x n → p} .

Thus S is closed if and only if S = lim S. Secondly, it's also possible for a set to be both open and closed; this will be discussed in Chapter 7.

The reason for the opposing terms is the following theorem:

Theorem 2.7.7 (Closed sets are complements of open sets)

Let M be a metric space, and S ⊆ M any subset. Then the following are equivalent:

• The set S is closed in M .

• The complement M \ S is open in M . 

Problem 2E.

Prove that a function f : R → R which is strictly increasing must be continuous at some point.

II Basic Abstract Algebra

Part II: Contents 3 Homomorphisms and quotient groups 3.1 Generators and group presentations .

3.3

Cosets and modding out . 

3.5

Eliminating the homomorphism .

3.6 (Digression)

The first isomorphism theorem .

3.7

A few harder problems to think about .

4

Rings and ideals 

4.3

Definition and examples of rings .

4.7

Generating ideals .

4.8

Principal ideal domains .

4.9

Noetherian rings . 

5

Flavors of rings 5.1

5.2

Integral domains .

5.3

Prime ideals .

5.4

Maximal ideals .

5.5

Field of fractions .

5.6

Unique factorization domains (UFD's) .

5.7

A few harder problems to think about .

3 Homomorphisms and quotient groups §3.1 Generators and group presentations

Prototypical example for this section:

D 2n = r, s | r n = s 2 = 1

Let G be a group. Recall that for some element x ∈ G, we could consider the subgroup . . . , x -2 , x -1 , 1, x, x 2 , . . .

of G.

Here's a more pictorial version of what we did: put x in a box, seal it tightly, and shake vigorously. Using just the element x, we get a pretty explosion that produces the subgroup above.

What happens if we put two elements x, y in the box? Among the elements that get produced are things like xyxyx, x 2 y 9 x -5 y 3 , y -2015 , . . . Essentially, I can create any finite product of x, y, x -1 , y -1 . This leads us to define: Definition 3.1. 1. Let S be a subset of G. The subgroup generated by S, denoted ⟨S⟩, is the set of elements which can be written as a finite product of elements in S (and their inverses). If ⟨S⟩ = G then we say S is a set of generators for G, as the elements of S together create all of G. Exercise 3.1.2. Why is the condition "and their inverses" not necessary if G is a finite group? (As usual, assume Lagrange's theorem.)

Example 3. 1.3 (Z is the infinite cyclic group) Consider 1 as an element of Z = (Z, +). We see ⟨1⟩ = Z, meaning {1} generates Z. It's important that -1, the inverse of 1 is also allowed: we need it to write all integers as the sum of 1 and -1.

This gives us an idea for a way to try and express groups compactly. Why not just write down a list of generators for the groups? For example, we could write Z ∼ = ⟨a⟩ meaning that Z is just the group generated by one element.

There's one issue: the generators usually satisfy certain properties. For example, consider Z/100Z. It's also generated by a single element x, but this x has the additional property that x 100 = 1. This motivates us to write Z/100Z = x | x 100 = 1 .

I'm sure you can see where this is going. All we have to do is specify a set of generators and relations between the generators, and say that two elements are equal if and only if you can get from one to the other using relations. Such an expression is appropriately called a group presentation. Example 3. 1 

.4 (Dihedral group)

The dihedral group of order 2n has a presentation

D 2n = r, s | r n = s 2 = 1, rs = sr -1 .

Thus each element of D 2n can be written uniquely in the form r α or sr α , where α = 0, 1, . . . , n -1.

Example 3. 1.5 (Klein four group)

The Klein four group, isomorphic to Z/2Z × Z/2Z, is given by the presentation The free group on n elements is the group whose presentation has n generators and no relations at all. It is denoted F n , so

F n = ⟨x 1 , x 2 , . . . , x n ⟩ .

In other words, F 2 = ⟨a, b⟩ is the set of strings formed by appending finitely many copies of a, b, a -1 , b -1 together.

Question 3. 1.7 . Notice that F 1 ∼ = Z.

Abuse of Notation 3. 1.8 . One might unfortunately notice that "subgroup generated by a and b" has exactly the same notation as the free group ⟨a, b⟩. We'll try to be clear based on context which one we mean.

Presentations are nice because they provide a compact way to write down groups. They do have some shortcomings, though. 1 Example 3. 1 

.9 (Presentations can look very different)

The same group can have very different presentations. For instance consider

D 2n = x, y | x 2 = y 2 = 1, (xy) n = 1.

. ( To see why this is equivalent, set x = s, y = rs.) §3.2 Homomorphisms Prototypical example for this section: The "mod out by 100" map, Z → Z/100Z.

How can groups talk to each other?

Two groups are "the same" if we can write an isomorphism between them. And as we saw, two metric spaces are "the same" if we can write a homeomorphism between them. But what's the group analogy of a continuous map? We simply drop the "bijection" condition.

Definition 3.2. 1 . Let G = (G, ⋆) and H = (H, * ) be groups. A group homomorphism is a map ϕ : G → H such that for any g 1 , g 2 ∈ G we have ϕ(g 1 ⋆ g 2 ) = ϕ(g 1 ) * ϕ(g 2 ).

(Not to be confused with "homeomorphism" from last chapter: note the spelling.)

Example 3.2.2 (Examples of homomorphisms)

Let G and H be groups.

(a) Any isomorphism G → H is a homomorphism. In particular, the identity map G → G is a homomorphism.

(b) The trivial homomorphism G → H sends everything to 1 H .

(c) There is a homomorphism from Z to Z/100Z by sending each integer to its residue modulo 100.

(d) There is a homomorphism from Z to itself by x → 10x which is injective but not surjective.

(e) There is a homomorphism from S n to S n+1 by "embedding": every permutation on {1, . . . , n} can be thought of as a permutation on {1, . . . , n + 1} if we simply let n + 1 be a fixed point.

(f) A homomorphism ϕ : D 12 → D 6 is given by s 12 → s 6 and r 12 → r 6 .

(g) Specifying a homomorphism Z → G is the same as specifying just the image of the element 1 ∈ Z. Why?

The last two examples illustrate something: suppose we have a presentation of G. To specify a homomorphism G → H, we only have to specify where each generator of G goes, in such a way that the relations are all satisfied. Important remark: the right way to think about an isomorphism is as a "bijective homomorphism". To be explicit, So the definitions of homeomorphism of metric spaces and isomorphism of groups are not too different.

Some obvious properties of homomorphisms follow. It is a subgroup of G (in particular, 1 G ∈ ker ϕ for obvious reasons).

Question 3.2.6. Verify that ker ϕ is in fact a subgroup of G.

We also have the following important fact, which we also encourage the reader to verify.

Proposition 3.2.7 (Kernel determines injectivity)

The map ϕ is injective if and only if ker ϕ = {1 G }.

To make this concrete, let's compute the kernel of each of our examples. (e) There is a homomorphism from S n to S n+1 by "embedding", but it also has trivial kernel because it is injective.

(f) A homomorphism ϕ : D 12 → D 6 is given by s 12 → s 6 and r 12 → r 6 . You can check that ker ϕ = 1, r 3 12 ∼ = Z/2Z.

(g) Exercise below.

Exercise 3.2.9. Fix any g ∈ G. Suppose we have a homomorphism Z → G by n → g n . What is the kernel? Question 3.2.10. Show that for any homomorphism ϕ : G → H, the image ϕ img (G) is a subgroup of H. Hence, we'll be especially interested in the case where ϕ is surjective.

§3.3 Cosets and modding out

Prototypical example for this section: Modding out by n: Z/(n • Z) ∼ = Z/nZ.

The next few sections are a bit dense. If this exposition doesn't work for you, try [Go11] .

Let G and Q be groups, and suppose there exists a surjective homomorphism ϕ : G ↠ Q.

In other words, if ϕ is injective then ϕ : G → Q is a bijection, and hence an isomorphism. But suppose we're not so lucky and ker ϕ is bigger than just {1 G }. What is the correct interpretation of a more general homomorphism? Let's look at the special case where ϕ : Z → Z/100Z is "modding out by 100". We already saw that the kernel of this map is ker ϕ = 100Z = {. . . , -200, -100, 0, 100, 200, . . . } .

Recall now that ker ϕ is a subgroup of G. What this means is that ϕ is indifferent to the subgroup 100Z of Z: ϕ(15) = ϕ(2000 + 15) = ϕ(-300 + 15) = ϕ(700 + 15) = . . . .

So Z/100Z

is what we get when we "mod out by 100". Cool .

In other words, let G be a group and ϕ : G ↠ Q be a surjective homomorphism with kernel N ⊆ G.

We claim that Q should be thought of as the quotient of G by N .

To formalize this, we will define a so-called quotient group G/N in terms of G and N only (without referencing Q) which will be naturally isomorphic to Q.

For motivation, let's give a concrete description of Q using just ϕ and G. The elements of each set all have the same image when we apply ϕ, and moreover any two elements in different sets have different images. Then the main idea is to notice that

We can think of Q as the group whose elements are the sets above.

Thus, given ϕ we define an equivalence relation ∼ N on G by saying x ∼ N y for ϕ(x) = ϕ(y). This ∼ N divides G into several equivalence classes in G which are in obvious bijection with Q, as above. Now we claim that we can write these equivalence classes very explicitly.

Exercise 3.3.1. Show that x ∼ N y if and only if x = yn for some n ∈ N (in the mod 100 example, this means they "differ by some multiple of 100"). Thus for any g ∈ G, the equivalence class of ∼ N which contains g is given explicitly by

gN := {gn | n ∈ N } .

Here's the word that describes the types of sets we're running into now. Definition 3.3.2. Let H be any subgroup of G (not necessarily the kernel of some homomorphism). A set of the form gH is called a left coset of H. Remark 3.3.3 -Although the notation might not suggest it, keep in mind that g 1 N is often equal to g 2 N even if g 1 ̸ = g 2 . In the "mod 100" example, 3 + N = 103 + N . In other words, these cosets are sets.

This means that if I write "let gH be a coset" without telling you what g is, you can't figure out which g I chose from just the coset itself. If you don't believe me, here's an example of what I mean:

x + 100Z = {. . . , -97, 3, 103, 203, . . . } =⇒ x = ?.

There's no reason to think I picked x = 3. (I actually picked x = -13597.) Remark 3.3.4 -Given cosets g 1 H and g 2 H, you can check that the map x → g 2 g - 1 1 x is a bijection between them. So actually, all cosets have the same cardinality.

So, long story short, Elements of the group Q are naturally identified with left cosets of N .

In practice, people often still prefer to picture elements of Q as single points (for example it's easier to think of Z/2Z as {0, 1} rather than {. . . , -2, 0, 2, . . . }, {. . . , -1, 1, 3, . . . } ).

If you like this picture, then you might then draw G as a bunch of equally tall fibers (the cosets), which are then "collapsed" onto Q.

⇐= ⇐= ⇐= ⇐= ⇐= ⇐=

G Q Now that we've done this, we can give an intrinsic definition for the quotient group we alluded to earlier.

Definition 3.3.5. A subgroup N of G is called normal if it is the kernel of some homomorphism. We write this as N ⊴ G. Definition 3.3.6. Let N ⊴ G. Then the quotient group, denoted G/N (and read "G mod N "), is the group defined as follows.

• The elements of G/N will be the left cosets of N .

• We want to define the product of two cosets C 1 and C 2 in G/N . Recall that the cosets are in bijection with elements of Q. So let q 1 be the value associated to the coset C 1 , and q 2 the one for C 2 . Then we can take the product to be the coset corresponding to q 1 q 2 .

Quite importantly, we can also do this in terms of representatives of the cosets. Let g 1 ∈ C 1 and g 2 ∈ C 2 , so C 1 = g 1 N and C 2 = g 2 N . Then C 1 • C 2 should be the coset which contains g 1 g 2 . This is the same as the above definition since ϕ(g 1 g 2 ) = ϕ(g 1 )ϕ(g 2 ) = q 1 q 2 ; all we've done is define the product in terms of elements of G, rather than values in H.

Using the gN notation, and with Remark 3. 3.3 in mind, we can write this even more succinctly: (g 1 N ) • (g 2 N ) := (g 1 g 2 )N.

And now you know why the integers modulo n are often written Z/nZ! Question 3.3.7. Take a moment to digest the above definition.

By the way we've built it, the resulting group G/N is isomorphic to Q. In a sense we think of G/N as "G modulo the condition that n = 1 for all n ∈ N ".

§3.4 (Optional) Proof of Lagrange's theorem

As an aside, with the language of cosets we can now show Lagrange's theorem in the general case.

Theorem 3.4.1 (Lagrange's theorem)

Let G be a finite group, and let H be any subgroup. Then |H| divides |G|.

The proof is very simple: note that the cosets of H all have the same size and form a partition of G (even when H is not necessarily normal). Hence if n is the number of cosets, then n • |H| = |G|. Let's look at the last definition of G/N we provided. The short version is:

• The elements of G/N are cosets gN , which you can think of as equivalence classes of a relation ∼ N (where g 1 ∼ N g 2 if g 1 = g 2 n for some n ∈ N ).

• Given cosets g 1 N and g 2 N the group operation is

g 1 N • g 2 N := (g 1 g 2 )N.

Question: where do we actually use the fact that N is normal? We don't talk about ϕ or Q anywhere in this definition.

The answer is in Remark 3.3.3. The group operation takes in two cosets, so it doesn't know what g 1 and g 2 are. But behind the scenes, the normal condition guarantees that the group operation can pick any g 1 and g 2 it wants and still end up with the same coset. If we didn't have this property, then it would be hard to define the product of two cosets C 1 and C 2 because it might make a difference which g 1 ∈ C 1 and g 2 ∈ C 2 we picked. The fact that N came from a homomorphism meant we could pick any representatives g 1 and g 2 of the cosets we wanted, because they all had the same ϕ-value.

We want some conditions which force this to be true without referencing ϕ at all. Suppose ϕ : G → K is a homomorphism of groups with H = ker ϕ. Aside from the fact H is a group, we can get an "obvious" property: Question 3.5. 1 . Show that if h ∈ H, g ∈ G, then ghg -1 ∈ H. (Check ϕ(ghg -1 ) = 1 K .)

Example 3.5.2 (Example of a non-normal subgroup)

Let D 12 = r, s | r 6 = s 2 = 1, rs = sr -1 . Consider the subgroup of order two H = {1, s} and notice that rsr -1 = r(sr -1 ) = r(rs) = r 2 s / ∈ H.

Hence H is not normal, and cannot be the kernel of any homomorphism.

Well, duh -so what? Amazingly it turns out that this is the sufficient condition we want. Specifically, it makes the nice "coset multiplication" we wanted work out.

Remark 3.5.3 (For math contest enthusiasts) -This coincidence is really a lot like functional equations at the IMO. We all know that normal subgroups H satisfy ghg -1 ∈ H; the surprise is that from the latter seemingly weaker condition, we can deduce H is normal.

Thus we have a new criterion for "normal" subgroups which does not make any external references to ϕ.

Theorem 3.5.4 (Algebraic condition for normal subgroups)

Let H be a subgroup of G. Then the following are equivalent:

• H ⊴ G.

• For every g ∈ G and h ∈ H, ghg -1 ∈ H. Proof . We already showed one direction.

For the other direction, we need to build a homomorphism with kernel H. So we simply define the group G/H as the cosets. To put a group operation, we need to verify:

Claim 3.5.5. If g ′ 1 ∼ H g 1 and g ′ 2 ∼ H g 2 then g ′ 1 g ′ 2 ∼ H g 1 g 2 .

Proof. Boring algebraic manipulation (again functional equation style). Let g ′ 1 = g 1 h 1 and g ′ 2 = g 2 h 2 , so we want to show that g 1 h 1 g 2 h 2 ∼ H g 1 g 2 . Since H has the property, g -1 2 h 1 g 2 is some element of H, say h 3 . Thus h 1 g 2 = g 2 h 3 , and the left-hand side becomes g 1 g 2 (h 3 h 2 ), which is fine since h 3 h 2 ∈ H.

■

With that settled we can just define the product of two cosets (of normal subgroups) by (g 1 H) • (g 2 H) = (g 1 g 2 )H.

Thus the claim above shows that this multiplication is well-defined (this verification is the "content" of the theorem). So G/H is indeed a group! Moreover there is an obvious "projection" homomorphism G → G/H (with kernel H), by g → gH.

Example 3.5.6 (Modding out in the product group) Consider again the product group G × H. Earlier we identified a subgroup

G ′ = {(g, 1 H ) | g ∈ G} ∼ = G.

You can easily see that G ′ ⊴ G × H. (Easy calculation.)

Moreover, you can check that

(G × H)/(G ′ ) ∼ = H.

Indeed, we have (g, h) ∼ G ′ (1 G , h) for all g ∈ G and h ∈ H. The kernel of this map is N = {1, r 2 , sr, sr 3 }.

We can do a quick computation of all the elements of D 8 to get ϕ(1) = ϕ(r 2 ) = ϕ(sr) = ϕ(sr 3 ) = 0 and ϕ(r) = ϕ(r 3 ) = ϕ(s) = ϕ(sr 2 ) = 2.

The two relevant fibers are ϕ pre (0) = 1N = r 2 N = srN = sr 3 N = {1, r 2 , sr, sr 3 } and ϕ pre (2) = rN = r 3 N = sN = sr 2 N = {r, r 3 , s, sr 2 }.

So we see that |D 8 /N | = 2 is a group of order two, or Z/2Z. Indeed, the image of ϕ is 0, 2 ∼ = Z/2Z.

Question 3.5.9. Suppose G is abelian. Why does it follow that any subgroup of G is normal?

Finally here's some food for thought: suppose one has a group presentation for a group G that uses n generators. Can you write it as a quotient of the form F n /N , where N is a normal subgroup of F n ? §3. 6 

(Digression) The first isomorphism theorem

One quick word about what other sources usually say.

Most textbooks actually define normal using the ghg -1 ∈ H property. Then they define G/H for normal H in the way I did above, using the coset definition

(g 1 H) • (g 2 H) = g 1 g 2 H.

Using purely algebraic manipulations (like I did) this is well-defined, and so now you have this group G/H or something. The underlying homomorphism isn't mentioned at all, or is just mentioned in passing.

I think this is incredibly dumb. The normal condition looks like it gets pulled out of thin air and no one has any clue what's going on, because no one has any clue what a normal subgroup actually should look like.

Other sources like to also write the so-called first isomorphism theorem. 2 It goes like this.

Theorem 3.6. 1 

(First isomorphism theorem)

Let ϕ : G → H be a homomorphism. Then G/ ker ϕ is isomorphic to ϕ img (G).

To me, this is just a clumsier way of stating the same idea.

About the only merit this claim has is that if ϕ is injective, then the image ϕ img (G) is an isomorphic copy of G inside the group H. (Try to see this directly!) This is a pattern we'll often see in other branches of mathematics: whenever we have an injective structure-preserving map, often the image of this map will be some "copy" of G. (Here "structure" refers to the group multiplication, but we'll see some more other examples of "types of objects" later!)

In that sense an injective homomorphism ϕ : G → H is an embedding of G into H. §3. Determine the order of G.

Problem 3H (Homophony group). The homophony group (of English) is the group with 26 generators a, b, . . . , z and one relation for every pair of English words which sound the same. For example knight = night (and hence k = 1). Prove that the group is trivial.

4 Rings and ideals §4.1 Some motivational metaphors about rings vs groups

In this chapter we'll introduce the notion of a commutative ring R. It is a larger structure than a group: it will have two operations addition and multiplication, rather than just one. We will then immediately define a ring homomorphism R → S between pairs of rings. This time, instead of having normal subgroups H ⊴ G, rings will instead have subsets I ⊆ R called ideals, which are not themselves rings but satisfy some niceness conditions. We will then show you how to define R/I, in analogy to G/H as before. Finally, like with groups, we will talk a bit about how to generate ideals.

Here is a possibly helpful I wrote most of these examples with a number theoretic eye in mind; thus if you liked elementary number theory, a lot of your intuition will carry over. Basically, we'll try to generalize properties of the ring Z to any abelian structure in which we can also multiply. That's why, for example, you can talk about "irreducible polynomials in Q[x]" in the same way you can talk about "primes in Z", or about "factoring polynomials modulo p" in the same way we can talk "unique factorization in Z". Even if you only care about Z (say, you're a number theorist), this has a lot of value: I assure you that trying to solve x n + y n = z n (for n > 2) requires going into a ring other than Z! Thus for all the sections that follow, keep Z in mind as your prototype. I mention this here because commutative algebra is also closely tied to algebraic geometry. Lots of the ideas in commutative algebra have nice "geometric" interpretations that motivate the definitions, and these connections are explored in the corresponding part later. So, I want to admit outright that this is not the only good way (perhaps not even the most natural one) of motivating what is to follow.

§4.3 Definition and examples of rings

Prototypical example for this section: Z all the way! Also R[x] and various fields (next section).

Well, I guess I'll define a ring 1 .

Definition 4.3.1.

A ring is a triple (R, +, ×), the two operations usually called addition and multiplication, such that (i) (R, +) is an abelian group, with identity 0 R , or just 0.

(ii) × is an associative, binary operation on R with some identity, written 1 R or just 1.

(iii) Multiplication distributes over addition.

The ring R is commutative if × is commutative.

Abuse of Notation 4.3.2. As usual, we will abbreviate (R, +, ×) to R.

Abuse of Notation 4.3.3. For simplicity, assume all rings are commutative for the rest of this chapter. We'll run into some noncommutative rings eventually, but for such rings we won't need the full theory of this chapter anyways. We also denote it by Z/nZ.

Here is also a trivial example.

Definition 4.3.5. The zero ring is the ring R with a single element. We denote the zero ring by 0. A ring is nontrivial if it is not the zero ring.

Exercise 4.3.6 (Comedic). Show that a ring is nontrivial if and only if 0 R ̸ = 1 R .

Since I've defined this structure, I may as well state the obligatory facts about it.

Fact 4.3.7. For any ring R and r ∈ R, r

• 0 R = 0 R . Moreover, r • (-1 R ) = -r.

Here are some more examples of rings.

Example 4.3.8 (Product ring)

Given two rings R and S the product ring, denoted R × S, is defined as ordered pairs (r, s) with both operations done component-wise. For example, the Chinese remainder theorem says that

Z/15Z ∼ = Z/3Z × Z/5Z

with the isomorphism n mod 15 → (n mod 3, n mod 5).

Remark 4.3.9 -Equivalently, we can define R × S as the abelian group R ⊕ S, and endow it with the multiplication where r • s = 0 for r ∈ R, s ∈ S. Given any ring R, the polynomial ring R[x] is defined as the set of polynomials with coefficients in R:

R[x] = a n x n + a n-1 x n-1 + • • • + a 0 | a 0 , . . . , a n ∈ R .

This is pronounced "R adjoin x". Addition and multiplication are done exactly in the way you would expect.

Remark 4.3.12 (Digression on division) -Happily, polynomial division also does what we expect: if p ∈ R[x] is a polynomial, and p(a) = 0, then (x -a)q(x) = p(x) for some polynomial q. Proof: do polynomial long division.

With that, note the caveat that

x 2 -1 ≡ (x -1)(x + 1) (mod 8) has four roots 1, 3, 5, 7 in Z/8Z.

The problem is that 2 • 4 = 0 even though 2 and 4 are not zero; we call 2 and 4 zero divisors for that reason. In an integral domain (a ring without zero divisors), this pathology goes away, and just about everything you know about polynomials carries over. (I'll say this all again next section.) The Gaussian integers are the set of complex numbers with integer real and imaginary parts, that is

Z[i] = {a + bi | a, b ∈ Z} .

Abuse of Notation 4.3.15 (Liberal use of adjoinment). Careful readers will detect some abuse in notation here. Z[i] should officially be "integer-coefficient polynomials in a variable i". However, it is understood from context that i 2 = -1; and a polynomial in i = √ -1 "is" a Gaussian integer.

Example 4.3.16 (Cube root of 2)

As another example (using the same abuse of notation):

Z[ 3 √ 2] = a + b 3 √ 2 + c 3 √ 4 | a, b, c ∈ Z . §4.

4 Fields

Prototypical example for this section: Q is a field, but Z is not.

Although we won't need to know what a field is until next chapter, they're so convenient for examples I will go ahead and introduce them now.

As you might already know, if the multiplication is invertible, then we call the ring a field. To be explicit, let me write the relevant definitions. (b) On the other hand, in Q everything is a unit (except 0). For example, 3 5 is a unit since 3 5 • 5 3 = 1.

(c) The Gaussian integers Z[i] have four units: ±1 and ±i.

Definition 4.4.3. A nontrivial (commutative) ring is a field when all its nonzero elements are units.

Colloquially, we say that

A field is a structure where you can add, subtract, multiply, and divide.

Depending on context, they are often denoted either k, K, F . (b) If p is a prime, then Z/pZ is a field, which we usually denote by F p .

The trivial ring 0 is not considered a field, since we require fields to be nontrivial.

§4.5 Homomorphisms

Prototypical example for this section: Z → Z/5Z by modding out by 5.

This section is going to go briskly -it's the obvious generalization of all the stuff we did with quotient groups. 2First, we define a homomorphism and isomorphism. 

(i) ϕ(x + R y) = ϕ(x) + S ϕ(y) for each x, y ∈ R. (ii) ϕ(x × R y) = ϕ(x) × S ϕ(y) for each x, y ∈ R. (iii) ϕ(1 R ) = 1 S .

If ϕ is a bijection then ϕ is an isomorphism and we say that rings R and S are isomorphic.

Just what you would expect. The only surprise is that we also demand ϕ(1 R ) to go to 1 S . This condition is not extraneous: consider the map Z → Z called "multiply by zero". Because we require 1 R to 1 S , some maps that you might have thought were homomorphisms will fail.

(a) The map Z → Z by x → 2x is not a ring homomorphism. Aside from the fact it sends 1 to 2, it also does not preserve multiplication.

(b) If S is a nontrivial ring, the map R → S by x → 0 is not a ring homomorphism, even though it preserves multiplication.

(c) There is no ring homomorphism Z/2016Z → Z at all.

In particular, whereas for groups G and H there was always a trivial group homomorphism sending everything in G to 1 H , this is not the case for rings.

§4.6 Ideals

Prototypical example for this section: The multiples of 5 are an ideal of Z.

Now, just like we were able to mod out by groups, we'd also like to define quotient rings. So once again, Definition 4.6.1. The kernel of a ring homomorphism ϕ : R → S, denoted ker ϕ, is the set of r ∈ R such that ϕ(r) = 0.

In group theory, we were able to characterize the "normal" subgroups by a few obviously necessary conditions (namely, gHg -1 = H). We can do the same thing for rings, and it's in fact easier because our operations are commutative.

First, note two obvious facts:

• If ϕ(x) = ϕ(y) = 0, then ϕ(x + y) = 0 as well. So ker ϕ should be closed under addition.

• If ϕ(x) = 0, then for any r ∈ R we have ϕ(rx) = ϕ(r)ϕ(x) = 0 too. So for x ∈ ker ϕ and any r ∈ R, we have rx ∈ ker ϕ.

A (nonempty) subset I ⊆ R is called an ideal if it satisfies these properties. That is, Definition 4.6.2. A nonempty subset I ⊆ R is an ideal if it is closed under addition, and for each x ∈ I, rx ∈ I for all r ∈ R. It is

proper if I ̸ = R.

Note that in the second condition, r need not be in I! So this is stronger than merely saying I is closed under multiplication.

Remark 4.6.3 -If R is not commutative, we also need the condition xr ∈ I. That is, the ideal is two-sided: it absorbs multiplication from both the left and the right. But since rings in Napkin are commutative we needn't worry with this distinction.

Example 4.6.4 (Prototypical example of an ideal) Consider the set I = 5Z = {. . . , -10, -5, 0, 5, 10, . . . } as an ideal in Z. We indeed see I is the kernel of the "take mod 5" homomorphism:

Z ↠ Z/5Z.

It's clearly closed under addition, but it absorbs multiplication from all elements of Z: given 15 ∈ I, 999 ∈ Z, we get 15 • 999 ∈ I. Let R be a ring and I ⊊ R. Then I is the kernel of some homomorphism if and only if it's an ideal.

Proof. It's quite similar to the proof for the normal subgroup thing, and you might try it yourself as an exercise.

Obviously the conditions are necessary. To see they're sufficient, we define a ring by "cosets"

S = {r + I | r ∈ R} .

These are the equivalence classes under r 1 ∼ r 2 if and only if r 1 -r 2 ∈ I (think of this as taking "mod I"). To see that these form a ring, we have to check that the addition and multiplication we put on them is well-defined. Specifically, we want to check that if r 1 ∼ s 1 and r 2 ∼ s 2 , then r 1 + r 2 ∼ s 1 + s 2 and r 1 r 2 ∼ s 1 s 2 . We actually already did the first part -just think of R and S as abelian groups, forgetting for the moment that we can multiply. The multiplication is more interesting.

Exercise 4.6.7 (Recommended). Show that if r 1 ∼ s 1 and r 2 ∼ s 2 , then r 1 r 2 ∼ s 1 s 2 . You will need to use the fact that I absorbs multiplication from any elements of R, not just those in I.

Anyways, since this addition and multiplication is well-defined there is now a surjective homomorphism R → S with kernel exactly I.

Definition 4.6.8. Given an ideal I, we define as above the quotient ring

R/I := {r + I | r ∈ R} .

It's the ring of these equivalence classes. This ring is pronounced "R mod I".

Example 4.6.9 (Z/5Z)

The integers modulo 5 formed by "modding out additively by 5" are the Z/5Z we have already met.

But here's an important point: just as we don't actually think of Z/5Z as consisting of k + 5Z for k = 0, . . . , 4, we also don't really want to think about R/I as elements r + I.

The better way to think about it is R/I is the result when we declare that elements of I are all zero; that is, we "mod out by elements of I".

For example, modding out by 5Z means that we consider all elements in Z divisible by 5 to be zero. 

§4.7 Generating ideals

Prototypical example for this section: In Z, the ideals are all of the form (n).

Let's give you some practice with ideals. An important piece of intuition is that once an ideal contains a unit, it contains 1, and thus must contain the entire ring. That's why the notion of "proper ideal" is useful language. To expand on that: Proposition 4.7.1 (Proper ideal ⇐⇒ no units)

Let R be a ring and I ⊆ R an ideal. Then I is proper (i.e. I ̸ = R) if and only if it contains no units of R.

Proof. Suppose I contains a unit u, i.e. an element u with an inverse u -1 . Then it contains u • u -1 = 1, and thus I = R. Conversely, if I contains no units, it is obviously proper.

As a consequence, if K is a field, then its only ideals are (0) and K (this was Exercise 4.6.5). So for our practice purposes, we'll be working with rings that aren't fields.

First practice: Z. Thus, while ideals of fields are not terribly interesting, ideals of Z look eerily like elements of Z. Let's make this more precise.

Definition 4.7.3. Let R be a ring. The ideal generated by a set of elements x 1 , . . . , x n ∈ R is denoted by I = (x 1 , x 2 , . . . , x n ) and given by

I = {r 1 x 1 + • • • + r n x n | r i ∈ R} .

One can think of this as "the smallest ideal containing all the x i ".

The analogy of putting the {x i } in a sealed box and shaking vigorously kind of works here too.

Remark 4.7.4 (Linear algebra digression) -If you know linear algebra, you can summarize this as: an ideal is an R-module. The ideal (x 1 , . . . , x n ) is the submodule spanned by x 1 , . . . , x n .

In particular, if I = (x) then I consists of exactly the "multiples of x", i.e. numbers of the form rx for r ∈ R.

Remark 4.7.5 -We can also apply this definition to infinite generating sets, as long as only finitely many of the r i are not zero (since infinite sums don't make sense in general). (c) Let R = Z[x] again and now let I = (x -3). Then R/I should be thought of as the quotient when x -3 ≡ 0, that is, x ≡ 3. So given a polynomial p(x) its image after we mod out should be thought of as p(3). Again R/I ∼ = Z, but in a different way.

(d) Finally, let I = (x -3, 5). Then R/I not only sends x to three, but also 5 to zero. So given p ∈ R, we get p(3) (mod 5). Then R/I ∼ = Z/5Z.

Remark 4.7.9 (Mod notation) -By the way, given an ideal I of a ring R, it's totally legit to write x ≡ y (mod I)

to mean that x -y ∈ I. Everything you learned about modular arithmetic carries over.

§4.8 Principal ideal domains

Prototypical example for this section:

Z is a PID, Z[x] is not. C[x] is a PID, C[x, y] is not.

What happens if we put multiple generators in an ideal, like (10, 15) ⊆ Z? Well, we have by definition that (10, 15) is given as a set by We call such an ideal (one generated by a single element) a principal ideal. So, in Z, every ideal is principal. But the same is not true in more general rings.

Example 4.8.1 (A non-principal ideal)

In Z[x], I = (x, 2015) is not a principal ideal.

For if I = (f ) for some polynomial f ∈ I then f divides x and 2015. This can only occur if f = ±1, but then I contains ±1, which it does not.

A ring with the property that all its ideals are principal is called a principal ideal ring. We like this property because they effectively let us take the "greatest common factor" in a similar way as the GCD in Z.

In practice, we actually usually care about so-called principal ideal domains (PID's). To reiterate, for now you should just verify that these are principal ideal rings, even though we are using the word PID.

(a) As we saw, Z is a PID.

(b) As we also saw, Z[x] is not a PID, since I = (x, 2015) for example is not principal.

(c) It turns out that for a field k the ring k[x] is always a PID. For example, Q

[x], R[x], C[x] are PID's.

If you want to try and prove this, first prove an analog of Bézout's lemma, which implies the result.

(d) C[x, y] is not a PID, because (x, y) is not principal. §4.9 Noetherian rings Prototypical example for this section: If it's too much to ask that an ideal is generated by one element, perhaps we can at least ask that our ideals are generated by finitely many elements. Unfortunately, in certain weird rings this is also not the case.

Z[x 1 , x 2 , .

Example 4.9.1 (Non-Noetherian ring) Consider the ring R = Z[x 1 , x 2 , x 3 , . . . ] which has infinitely many free variables. Then the ideal I = (x 1 , x 2 , . . . ) ⊆ R cannot be written with a finite generating set. Nonetheless, most "sane" rings we work in do have the property that their ideals are finitely generated. We now name such rings and give two equivalent definitions: Proposition 4.9.2 (The equivalent definitions of a Noetherian ring) For a ring R, the following are equivalent:

(a) Every ideal I of R is finitely generated (i.e. can be written with a finite generating set).

(b) There does not exist an infinite ascending chain of ideals

I 1 ⊊ I 2 ⊊ I 3 ⊊ . . . .

The absence of such chains is often called the ascending chain condition.

Such rings are called Noetherian.

Example 4.9.3 (Non-Noetherian ring breaks ACC)

In the ring R = Z[x 1 , x 2 , x 3 , . . . ] we have an infinite ascending chain

(x 1 ) ⊊ (x 1 , x 2 ) ⊊ (x 1 , x 2 , x 3 ) ⊊ . . . .

From the example, you can kind of see why the proposition is true: from an infinitely generated ideal you can extract an ascending chain by throwing elements in one at a This leaves the question: is our prototypical non-example of a PID, Z[x], a Noetherian ring? The answer is a glorious yes, according to the celebrated Hilbert basis theorem.

Theorem 4.9.5 (Hilbert basis theorem) Given a Noetherian ring R, the ring R[x] is also Noetherian. Thus by induction, R[x 1 , x 2 , . . . , x n ] is Noetherian for any integer n.

The proof of this theorem is really olympiad flavored, so I couldn't possibly spoil it -I've left it as a problem at the end of this chapter.

Noetherian rings really shine in algebraic geometry, and it's a bit hard for me to motivate them right now, other than to say "most rings you'll encounter are Noetherian". Please bear with me! §4.10 A few harder problems to think about 

Problem 4A. The ring R = R[x]/(x 2 + 1) is one that you've seen before. What is its name? Problem 4B. Show that C[x]/(x 2 -x) ∼ = C × C.

F p [x] → F p [x] by Ψ n i=0 a i x i = n i=0 a i x p i .

Let S denote the image of Ψ.

(a) Show that S is a ring with addition given by polynomial addition, and multiplication given by function composition.

(b) Prove that Ψ :

F p [x] → S is then a ring isomorphism.

We continue our exploration of rings by considering some nice-ness properties that rings or ideals can satisfy, which will be valuable later on. As before, number theory is interlaced as motivation We already saw this definition last chapter: a field K is a nontrivial ring for which every nonzero element is a unit.

In particular, there are only two ideals in a field: the ideal (0), which is maximal, and the entire field K.

§5.2 Integral domains

Prototypical example for this section: Z is an integral domain.

In practice, we are often not so lucky that we have a full-fledged field. Now it would be nice if we could still conclude the zero product property: if ab = 0 then either a = 0 or b = 0. If our ring is a field, this is true: if b ̸ = 0, then we can multiply by b -1 to get a = 0. But many other rings we consider like Z and Z[x] also have this property, despite not having division.

Not all rings though: in Z/15Z, Every field is an integral domain, so all the previous examples apply. In addition:

3 • 5 ≡ 0 (mod 15).

(a) Z is an integral domain, but it is not a field.

(b) R[x] is not a field, since there is no polynomial P (x) with xP (x) = 1. However, R[x] is an integral domain, because if P (x)Q(x) = 0 then one of P or Q is zero.

(c) Z[x] is also an example of an integral domain. In fact, R[x] is an integral domain for any integral domain R (why?).

(d) Z/nZ is a field (hence integral domain) exactly when n is prime. When n is not prime, it is a ring but not an integral domain.

The trivial ring 0 is not considered an integral domain.

At this point, we go ahead and say:

Definition 5.2.5. An integral domain where all ideals are principal is called a principal ideal domain (PID).

The ring Z/6Z is an example of a ring which is a principal ideal ring, but not an integral domain. As we alluded to earlier, we will never really use "principal ideal ring" in any real way: we typically will want to strengthen it to PID.

§5.3 Prime ideals

Prototypical example for this section: (5) is a prime ideal of Z.

We know that every integer can be factored (up to sign) as a unique product of primes; for example 15 = 3 • 5 and -10 = -2 • 5. You might remember the proof involves the so-called Bézout's lemma, which essentially says that (a, b) = (gcd(a, b)); in other words we've carefully used the fact that Z is a PID.

It turns out that for general rings, the situation is not as nice as factoring elements because most rings are not PID's. The classic example of something going wrong is

6 = 2 • 3 = 1 - √ -5 1 + √ -5 in Z[ √ -5].

Nonetheless, we can sidestep the issue and talk about factoring ideals: somehow the example 10 = 2 • 5 should be (10) = (2) • (5), which says "every multiple of 10 is the product of a multiple of 2 and a multiple of 5". The condition that I is proper is analogous to the fact that we don't consider 1 to be a prime number. 

(d) The ideal (x 2 ) of Z[x] is not prime, since x • x = x 2 .

(e) The ideal (3, x) of Z[x] is prime. This is actually easiest to see using Theorem 5. 3.5 below.

(f) The ideal (5) = 5Z + 5iZ of Z[i] is not prime, since the elements 3 + i and 3 -i have product 10 ∈ (5), yet neither is itself in (5).

Remark 5.3.3 -Ideals have the nice property that they get rid of "sign issues". For example, in Z, do we consider -3 to be a prime? When phrased with ideals, this annoyance goes away: (-3) = (3). More generally, for a ring R, talking about ideals lets us ignore multiplication by a unit. (Note that -1 is a unit in Z.)

Exercise 5.3.4. What do you call a ring R for which the zero ideal (0) is prime?

We also have:

Theorem 5.3.5 (Prime ideal ⇐⇒ quotient is integral domain)

An ideal I is prime if and only if R/I is an integral domain. I now must regrettably inform you that unique factorization is still not true even with the notion of a "prime" ideal (though again I haven't told you how to multiply two ideals yet). But it will become true with some additional assumptions that will arise in algebraic number theory (relevant buzzword: Dedekind domain).

§5.4 Maximal ideals

Prototypical example for this section: The ideal (x, 5) is maximal in Z[x], by quotient-ing.

Here's another flavor of an ideal. There's an analogous theorem to the one for prime ideals. If I is a maximal ideal of a ring R, then I is prime.

Proof. If I is maximal, then R/I is a field, hence an integral domain, so I is prime.

In practice, because modding out by generated ideals is pretty convenient, this is a very efficient way to check whether an ideal is maximal.

Example 5.4.6 (Modding out in Z[x])

(a) This instantly implies that (x, 5) is a maximal ideal in Z[x], because if we mod out by x and 5 in Z[x], we just get F 5 , which is a field.

(b) On the other hand, modding out by just x gives Z, which is an integral domain but not a field; that's why (x) is prime but not maximal.

As we saw, any maximal ideal is prime. But now note that Z has the special property that all of its nonzero prime ideals are also maximal. It's with this condition and a few other minor conditions that you get a so-called Dedekind domain where prime factorization of ideals does work. More on that later. §5.5 Field of fractions Prototypical example for this section: Frac(Z) = Q.

As long as we are here, we take the time to introduce a useful construction that turns any integral domain into a field. In fact everything you know about Q basically carries over by analogy. You can prove if you want that this indeed a field, but considering how comfortable we are that Q is well-defined, I wouldn't worry about it. . . 

R(x) = f (x) g(x) | f, g ∈ R[x] .

For example, 2x

x 2 -3 might be a typical element.

Example 5.5.4 (Gaussian rationals)

Just like we defined Z[i] by abusing notation, we can also write

Q(i) = Frac(Z[i]).

Officially, it should consist of

Q(i) = f (i) g(i) | g(i) ̸ = 0

for polynomials f and g with rational coefficients. But since i 2 = -1 this just leads to

Q(i) = a + bi c + di | a, b, c, d ∈ Q, (c, d) ̸ = (0, 0) .

And since

1 c+di = c-di c 2 +d 2 we end up with Q(i) = {a + bi | a, b ∈ Q} . §5.

6 Unique factorization domains (UFD's)

Prototypical example for this section: Z and polynomial rings in general.

Here is one stray definition that will be important for those with a number-theoretic inclination. Over the positive integers, we have a fundamental theorem of arithmetic, stating that every integer is uniquely the product of prime numbers.

We can even make an analogous statement in Z or Z[i], if we allow representations like 6 = (-2)(-3) and so on. The trick is that we only consider everything up to units; so 6 = (-2)(-3) = 2 • 3 are considered the same.

The general definition goes as follows.

Definition 5.6.1. A nonzero non-unit of an integral domain R is irreducible if it cannot be written as the product of two non-units.

An integral domain R is a unique factorization domain if every nonzero nonunit of R can be written as the product of irreducible elements, which is unique up to multiplication by units.

Question 5.6.2. Verify that Z is a UFD.

Example 5.6.3 (Examples of UFD's) (a) Fields are a "degenerate" example of UFD's: every nonzero element is a unit, so there is nothing to check.

(b) Z is a UFD. The irreducible elements are p and -p, for example 5 or -17.

(c) Q[x] is a UFD: polynomials with rational coefficients can be uniquely factored, up to scaling by constants (as the units of Q[x] are just the rational numbers).

(d) Z[x] is a UFD.

(e) The Gaussian integers Z[i] turns out to be a UFD too (and this will be proved in the chapters on algebraic number theory).

(f) Z[ √ -5] is the classic non-example of a UFD: one may write

6 = 2 • 3 = 1 - √ -5 1 + √ -5

but each of 2, 3, 1 ± √ -5 is irreducible. (It turns out the right way to fix this is by considering prime ideals instead, and this is one big motivation for Part XIII.) (g) Theorem we won't prove: every PID is a UFD.

(h) Theorem we won't prove: if R is a UFD, so is R[x] (and hence by induction so is R[x, y], R[x, y, z], . 

Q[ √ 2] = a + b √ 2 | a, b ∈ Q .

Is it a field? Problem 5B (Homomorphisms from fields are injective). Let K be a field and R a ring.

Prove that any homomorphism ψ :

K → R is injective. 2

Problem 5C ⋆ (Pre-image of prime ideals). Suppose ϕ : R → S is a ring homomorphism, and I ⊆ S is a prime ideal. Prove that ϕ pre (I) is prime as well.

Problem 5D ⋆ . Let R be an integral domain with finitely many elements. Prove that R is a field.

Problem 5E ⋆ (Krull's theorem). Let R be a ring and J a proper ideal.

(a) Prove that if R is Noetherian, then J is contained in a maximal ideal I. Problem 5G. Prove that any nonzero prime ideal of Z[ √ 2] is also a maximal ideal.

III Basic Topology

Part III: Contents 

6.3

Let the buyer beware .

6.4

Subspaces, and (inb4) a confusing linguistic point .

6.5

A few harder problems to think about .

7

Topological spaces 107

7.1

Forgetting the metric .

7.2

Re-definitions .

7.3

Hausdorff spaces .

7.5

Connected spaces .

7.6

Path-connected spaces .

7.7

Homotopy and simply connected spaces .

7.8

Bases of spaces .

7.9

A few harder problems to think about . 

8.2

Criteria for compactness .

8.3

Compactness using open covers .

8.4

Applications of compactness . 

8.6

A few harder problems to think about .

At the end of the last chapter on metric spaces, we introduced two adjectives "open" and "closed". These are important because they'll grow up to be the definition for a general topological space, once we graduate from metric spaces.

To move forward, we provide a couple niceness adjectives that applies to entire metric spaces, rather than just a set relative to a parent space. They are "(totally) bounded" and "complete". These adjectives are specific to metric spaces, but will grow up to become the notion of compactness, which is, in the words of [Pu02] , "the single most important concept in real analysis". At the end of the chapter, we will know enough to realize that something is amiss with our definition of homeomorphism, and this will serve as the starting point for the next chapter, when we define fully general topological spaces. §6.1 Boundedness Prototypical example for this section:

[0, 1] is bounded but R is not.

Here is one notion of how to prevent a metric space from being a bit too large.

Definition 6.1.1. A metric space M is bounded if there is a constant D such that d(p, q) ≤ D for all p, q ∈ M .

You can change the order of the quantifiers: Proposition 6. 1 

.2 (Boundedness with radii instead of diameters)

A metric space M is bounded if and only if for every point p ∈ M , there is a radius R (possibly depending on p) such that d(p, q) ≤ R for all q ∈ M . (d) A discrete space on an infinite set is bounded.

(e) N is not bounded, despite being homeomorphic to the discrete space! The fact that a discrete space on an infinite set is "bounded" might be upsetting to you, so here is a somewhat stronger condition you can use: Definition 6. 1.5 . A metric space is totally bounded if for any ε > 0, we can cover M with finitely many ε-neighborhoods.

For example, if ε = 1/2, you can cover [0, 1] 2 by ε-neighborhoods. 

(b) So for example [0, 1] or [0, 2] × [0, 3] is totally bounded.

(c) In contrast, a discrete space on an infinite set is not totally bounded.

§6.2 Completeness

Prototypical example for this section: R is complete, but Q and (0, 1) are not.

So far we can only talk about sequences converging if they have a limit. But consider the sequence

x 1 = 1, x 2 = 1.4, x 3 = 1.41, x 4 = 1.414, . . . .

It converges to √

2 in R, of course. But it fails to converge in Q; there is no rational number this sequence converges to. And so somehow, if we didn't know about the existence of R, we would have no idea that the sequence (x n ) is "approaching" something.

That seems to be a shame. Let's set up a new definition to describe these sequences whose terms get close to each other, even if they don't approach any particular point in the space. Thus, we only want to mention the given points in the definition. Definition 6.2.1. Let x 1 , x 2 , . . . be a sequence which lives in a metric space M = (M, d M ). We say the sequence is Cauchy if for any ε > 0, we have

d M (x m , x n ) < ε

for all sufficiently large m and n. The open interval (0, 1) is not complete, as the sequence 0.9, 0.99, 0.999, 0.9999, . . . is Cauchy but does not converge.

So, metric spaces need not be complete, like Q. But we certainly would like them to be complete, and in light of the following theorem this is not unreasonable.

Theorem 6.2.6 (Completion)

Every metric space can be "completed", i.e. made into a complete space by adding in some points.

We won't need this construction at all, so it's left as Problem 6C † .

Example 6.2.7 (Q completes to R)

The completion of Q is R.

(In fact, by using a modified definition of completion not depending on the real numbers, other authors often use this as the definition of R.)

§6.3 Let the buyer beware

There is something suspicious about both these notions: neither are preserved under homeomorphism! Example 6.3.1 (Something fishy is going on here) Let M = (0, 1) and N = R. As we saw much earlier M and N are homeomorphic. However:

• (0, 1) is totally bounded, but not complete.

• R is complete, but not bounded. This is the first hint of something going awry with the metric. As we progress further into our study of topology, we will see that in fact open sets and closed sets (which we motivated by using the metric) are the notion that will really shine later on. I insist on introducing the metric first so that the standard pictures of open sets and closed sets make sense, but eventually it becomes time to remove the training wheels. §6.4 Subspaces, and (inb4) a confusing linguistic point Prototypical example for this section: A circle is obtained as a subspace of R 2 .

As we've already been doing implicitly in examples, we'll now say: Definition 6.4.1. Every subset S ⊆ M is a metric space in its own right, by re-using the distance function on M . We say that S is a subspace of M .

For example, we saw that the circle S 1 is just a subspace of R 2 . It thus becomes important to distinguish between (i) "absolute" adjectives like "complete" or "bounded", which can be applied to both spaces, and hence even to subsets of spaces (by taking a subspace), and (ii) "relative" adjectives like "open (in M )" and "closed (in M )", which make sense only relative to a space, even though people are often sloppy and omit them.

So "[0, 1] is complete" makes sense, as does "[0, 1] is a complete subset of R", which we take to mean "[0, 1] is a complete subspace of R". This is since "complete" is an absolute adjective.

But here are some examples of ways in which relative adjectives require a little more care:

• Consider the sequence 1, 1.4, 1.41, 1.414, . . . . Viewed as a sequence in R, it converges to √ 2. But if viewed as a sequence in Q, this sequence does not converge! Similarly, the sequence 0.9, 0.99, 0.999, 0.9999 does not converge in the space (0, 1), although it does converge in [0, 1].

The fact that these sequences fail to converge even though they "ought to" is weird and bad, and was why we defined complete spaces to begin with.

• In general, it makes no sense to ask a question like "is [0, 1] open?". The questions "is [0, 1] open in R?" and "is [0, 1] open in [0, 1]?" do make sense, however. The answer to the first question is "no" but the answer to the second question is "yes"; indeed, every space is open in itself. Similarly, [0, 1 2 ) is an open set in the space M = [0, 1] because it is the ball in M of radius 1 2 centered at 0.

• Dually, it doesn't make sense to ask "is [0, 1] closed"? It is closed in R and in itself (but every space is closed in itself, anyways).

To make sure you understand the above, here are two exercises to help you practice relative adjectives. This illustrates a third point: a nontrivial set can be both open and closed. 1 As we'll see in Chapter 7, this implies the space is disconnected; i.e. the only examples look quite like the one we've given above. §6.5 A few harder problems to think about Problem 6A † (Banach fixed point theorem). Let M = (M, d) be a complete metric space. Suppose T : M → M is a continuous map such that for any p ̸ = q ∈ M , d (T (p), T (q)) < 0.999d(p, q).

(We call T a contraction.) Show that T has a unique fixed point.

Problem 6B (Henning Makholm, on math.SE). We let M and N denote the metric spaces obtained by equipping R with the following two metrics:

d M (x, y) = min {1, |x -y|} d N (x, y) = |e x -e y | .

(a) Fill in the following 2 × 3 table with "yes" or "no" for each cell. Problem 6E. Prove that Q is not homeomorphic to any complete metric space. 1 Which always gets made fun of.

7 Topological spaces

In Chapter 2 we introduced the notion of space by describing metrics on them. This gives you a lot of examples, and nice intuition, and tells you how you should draw pictures of open and closed sets.

However, moving forward, it will be useful to begin thinking about topological spaces in terms of just their open sets. (One motivation is that our fishy Example 6. 3 .1 shows that in some ways the notion of homeomorphism really wants to be phrased in terms of open sets, not in terms of the metric.) As we are going to see, the open sets manage to actually retain nearly all the information we need, but are simpler. 1 This will be done in just a few sections, and after that we will start describing more adjectives that we can apply to topological (and hence metric) spaces.

The most important topological notion is missing from this chapter: that of a compact space. It is so important that I have dedicated a separate chapter just for it.

Quick note for those who care: the adjectives "Hausdorff", "connected", and later "compact" are all absolute adjectives. §7.1 Forgetting the metric Recall Theorem 2.6.11:

A function f : M → N of metric spaces is continuous if and only if the pre-image of every open set in N is open in M .

Despite us having defined this in the context of metric spaces, this nicely doesn't refer to the metric at all, only the open sets. As alluded to at the start of this chapter, this is a great motivation for how we can forget about the fact that we had a metric to begin with, and rather start with the open sets instead.

Definition 7.1.1. A topological space is a pair (X, T ), where X is a set of points, and T is the topology, which consists of several subsets of X, called the open sets of X. The topology must obey the following axioms.

• ∅ and X are both in T . But by an "r-neighborhood" I always mean the points with distance less than r from x, and so I can only use this term if my space is a metric space.

§7.2 Re-definitions

Now that we've defined a topological space, for nearly all of our metric notions we can write down as the definition the one that required only open sets (which will of course agree with our old definitions when we have a metric space). §7. Therefore, any property defined only in terms of open sets is preserved by homeomorphism. Such a property is called a topological property. The later adjectives we define ("connected", "Hausdorff", "compact") will all be defined only in terms of open sets, so they will be topological properties.

§7.2.ii Closed sets

We saw last time there were two equivalent definitions for closed sets, but one of them relies only on open sets, and we use it: Definition 7.2.4. In a general topological space X, we say that

S ⊆ X is closed in X if the complement X \ S is open in X.

If S ⊆ X is any set, the closure of S, denoted S, is defined as the smallest closed set containing S. 

§7.2.iii Properties that don't carry over

Not everything works: Remark 7.2.7 (Complete and (totally) bounded are metric properties) -The two metric properties we have seen, "complete" and "(totally) bounded", are not topological properties. They rely on a metric, so as written we cannot apply them to topological spaces. One might hope that maybe, there is some alternate definition (like we saw for "continuous function") that is just open-set based. But Example 6.3.1 showing (0, 1) ∼ = R tells us that it is hopeless.

Remark 7.2.8 (Sequences don't work well) -You could also try to port over the notion of sequences and convergent sequences. However, this turns out to break a lot of desirable properties. Therefore I won't bother to do so, and thus if we are discussing sequences you should assume that we are working with a metric space.

§7.3 Hausdorff spaces

Prototypical example for this section: Every space that's not the Zariski topology (defined much later).

As you might have guessed, there exist topological spaces which cannot be realized as metric spaces (in other words, are not metrizable). One example is just to take X = {a, b, c} and the topology τ X = {∅, {a, b, c}}. This topology is fairly "stupid": it can't tell apart any of the points a, b, c! But any metric space can tell its points apart (because d(x, y) > 0 when x ̸ = y).

We'll see less trivial examples later, but for now we want to add a little more sanity condition onto our spaces. There is a whole hierarchy of such axioms, labelled T n for integers n (with n = 0 being the weakest and n = 6 the strongest); these axioms are called separation axioms.

By far the most common hypothesis is the T 2 axiom, which bears a special name.

Definition 7.3.1. A topological space X is Hausdorff if for any two distinct points p and q in X, there exists an open neighborhood U of p and an open neighborhood V of q such that U ∩ V = ∅.

In other words, around any two distinct points we should be able to draw disjoint open neighborhoods. Here's a picture to go with above, but not much going on.

p q Question 7.3.2. Show that all metric spaces are Hausdorff.

I just want to define this here so that I can use this word later. In any case, basically any space we will encounter other than the Zariski topology is Hausdorff. §7.4 Subspaces Prototypical example for this section: S 1 is a subspace of R 2 .

One can also take subspaces of general topological spaces. Needless to say, for metric spaces it doesn't matter which of these definitions I choose. (Proving this turns out to be surprisingly annoying, so I won't do so.) §7.5 Connected spaces Prototypical example for this section:

[0, 1] ∪ [2, 3] is disconnected.

Even in metric spaces, it is possible for a set to be both open and closed. For example ∅ and the entire space are examples of clopen sets. In fact, the presence of a nontrivial clopen set other than these two leads to a so-called disconnected space.

Question 7.5.2. Show that a space X has a nontrivial clopen set (one other than ∅ and X) if and only if X can be written as a disjoint union of two nonempty open sets.

We say X is disconnected if there are nontrivial clopen sets, and connected otherwise. To see why this should be a reasonable definition, it might help to solve Problem 7A † .

Example 7.5.3 (Disconnected and connected spaces) (a) The metric space

{(x, y) | x 2 + y 2 ≤ 1} ∪ {(x, y) | (x -4) 2 + y 2 ≤ 1} ⊆ R 2 is disconnected (it consists of two disks). (b) The space [0, 1] ∪ [2, 3

] is disconnected: it consists of two segments, each of which is a clopen set.

(c) A discrete space on more than one point is disconnected, since every set is clopen in the discrete space.

(d) Convince yourself that the set

x ∈ Q | x 2 < 2014 is a clopen subset of Q. Hence Q is disconnected too -it has gaps.

(e) [0, 1] is connected.

§7.6 Path-connected spaces

Prototypical example for this section: Walking around in C.

A stronger and perhaps more intuitive notion of a connected space is a path-connected space. The short description: "walk around in the space". Definition 7.6.1. A path in the space X is a continuous function

γ : [0, 1] → X.

Its endpoints are the two points γ(0) and γ(1).

You can think of [0, 1] as measuring "time", and so we'll often write γ(t) for t ∈ [0, 1] (with t standing for "time"). Here's a picture of a path.

X γ(0)

γ(1) γ Question 7.6.2. Why does this agree with your intuitive notion of what a "path" is? Definition 7.6.3. A space X is path-connected if any two points in it are connected by some path.

Exercise 7.6.4 (Path-connected implies connected). Let X = U ⊔ V be a disconnected space. Show that there is no path from a point of U to point

V . (If γ : [0, 1] → X, then we get [0, 1] = γ pre (U ) ⊔ γ pre (V ), but [0, 1] is connected.)

Example 7.6.5 (Examples of path-connected spaces)

• R 2 is path-connected, since we can "connect" any two points with a straight line.

• The unit circle S 1 is path-connected, since we can just draw the major or minor arc to connect two points. §7.7 Homotopy and simply connected spaces Prototypical example for this section: C and C \ {0}. Now let's motivate the idea of homotopy. Consider the example of the complex plane C (which you can think of just as R 2 ) with two points p and q. There's a whole bunch of paths from p to q but somehow they're not very different from one another. If I told you "walk from p to q" you wouldn't have too many questions.

C p q

So we're living happily in C until a meteor strikes the origin, blowing it out of existence. Then suddenly to get from p to q, people might tell you two different things: "go left around the meteor" or "go right around the meteor".

C \ {0} p q

So what's happening? In the first picture, the red, green, and blue paths somehow all looked the same: if you imagine them as pieces of elastic string pinned down at p and q, you can stretch each one to any other one.

But in the second picture, you can't move the red string to match with the blue string: there's a meteor in the way. The paths are actually different. 3The formal notion we'll use to capture this is homotopy equivalence. We want to write a definition such that in the first picture, the three paths are all homotopic, but the two paths in the second picture are somehow not homotopic. And the idea is just continuous deformation.

Definition 7.7.1. Let α and β be paths in X whose endpoints coincide. A (path) homotopy from α to β is a continuous function F : [0, 1] 2 → X, which we'll write F s (t) for s, t ∈ [0, 1], such that

F 0 (t) = α(t) and F 1 (t) = β(t) for all t ∈ [0, 1]

and moreover

α(0) = β(0) = F s (0) and α(1) = β(1) = F s (1) for all s ∈ [0, 1].

If a path homotopy exists, we say α and β are path homotopic and write α ≃ β.

Abuse of Notation 7.7.2. While I strictly should say "path homotopy" to describe this relation between two paths, I will shorten this to just "homotopy" instead. Similarly I will shorten "path homotopic" to "homotopic".

Animated picture: https://commons.wikimedia.org/wiki/File:HomotopySmall.gif. Needless to say, ≃ is an equivalence relation.

What this definition is doing is taking α and "continuously deforming" it to β, while keeping the endpoints fixed. Note that for each particular s, F s is itself a function. So s represents time as we deform α to β: it goes from 0 to 1, starting at α and ending at β. A space X is simply connected if it's path-connected and for any points p and q, all paths from p to q are homotopic.

That's why you don't ask questions when walking from p to q in C: there's really only one way to walk. Hence the term "simply" connected. But okay, this isn't that different. All I've done is taken a bunch of my prototypes and threw a bunch of ∪ signs at it. And that's the idea behind a basis. The open intervals form a basis of R.

In fact, more generally we have: Theorem 7.8.4 (Basis of metric spaces)

The r-neighborhoods form a basis of any metric space M .

Proof. Kind of silly -given an open set U draw an r p -neighborhood U p contained entirely inside U . Then p U p is contained in U and covers every point inside it.

Hence, an open set in R 2 is nothing more than a union of a bunch of open disks, and so on. The point is that in a metric space, the only open sets you really ever have to worry too much about are the r-neighborhoods. §7.9 A few harder problems to think about Problem 7A † . Let X be a topological space. Show that there exists a nonconstant continuous function X → {0, 1} if and only if X is disconnected (here {0, 1} is given the discrete topology). ⋆ . Let X and Y be topological spaces and let f : X → Y be a continuous function.

Problem 7B

(a) Show that if X is connected then so is f img (X).

(b) Show that if X is path-connected then so is f img (X).

Problem 7C (Hausdorff implies T 1 axiom). Let X be a Hausdorff topological space. Prove that for any point p ∈ X the set {p} is closed.

Problem 7D ([Pu02] , Exercise 2.56). Let M be a metric space with more than one point but at most countably infinitely many points. Show that M is disconnected. 4 Prove that there exists such a disjoint union with at most countably many intervals in it.

8 Compactness

One of the most important notions of topological spaces is that of compactness. It generalizes the notion of "closed and bounded" in Euclidean space to any topological space (e.g. see Problem 8F † ).

For metric spaces, there are two equivalent ways of formulating compactness:

• A "natural" definition using sequences, called sequential compactness.

• A less natural definition using open covers.

As I alluded to earlier, sequences in metric spaces are super nice, but sequences in general topological spaces suck (to the point where I didn't bother to define convergence of general sequences). So it's the second definition that will be used for general spaces.

§8.1 Definition of sequential compactness

Prototypical example for this section: [0, 1] is compact, but (0, 1) is not.

To emphasize, compactness is one of the best possible properties that a metric space can have. Definition 8.1.1. A subsequence of an infinite sequence x 1 , x 2 , . . . is exactly what it sounds like: a sequence x i 1 , x i 2 , . . . where i 1 < i 2 < • • • are positive integers. Note that the sequence is required to be infinite.

Another way to think about this is "selecting infinitely many terms" or "deleting some terms" of the sequence, depending on whether your glass is half empty or half full. 

S 2 = (x, y, z) | x 2 + y 2 + z 2 = 1 is compact. (c) The unit ball B 2 = (x, y) | x 2 + y 2 ≤ 1 is compact. (d)

The Hawaiian earring living in R 2 is compact: it consists of mutually tangent circles of radius If X and Y are compact spaces, then so is X × Y .

Proof. Problem 8E.

We also have:

Theorem 8.2.2 (The interval is compact) [0, 1] is compact.

Proof. Killed by Problem 8F † ; however, here is a sketch of a direct proof.

Split [0, 1] into [0, 1 2 ] ∪ [ 1 2 , 1]

. By Pigeonhole, infinitely many terms of the sequence lie in the left half (say); let x 1 be the first one and then keep only the terms in the left half after x 1 . Now split [0, 1 2 ] into [0, 

box [a 1 , b 1 ] × [a 2 , b 2 ] × • • • × [a n , b n ]. By Tychonoff's theorem, since each [a i , b i ] is compact the entire box is.

Since S is a closed subset of this compact box, we're done.

One really has to work in R n for this to be true! In other spaces, this criterion can easily fail.

Example 8.2.5 (Closed and bounded but not compact) Let S = {s 1 , s 2 , . . . } be any infinite set equipped with the discrete metric. Then S is closed (since all convergent sequences are constant sequences) and S is bounded (all points are a distance 1 from each other) but it's certainly not compact since the sequence s 1 , s 2 , . . . doesn't converge.

The Bolzano-Weierstrass theorem, which is Problem 8F † , tells you exactly which sets are compact in metric spaces in a geometric way.

§8.3 Compactness using open covers

Prototypical example for this section: [0, 1] is compact.

There's a second related notion of compactness which I'll now define. The following definitions might appear very unmotivated, but bear with me. Definition 8.3. 1. An open cover of a topological space X is a collection of open sets {U α } (possibly infinite or uncountable) which cover it: every point in X lies in at least one of the U α , so that X = U α .

Such a cover is called an open cover.

A subcover is exactly what it sounds like: it takes only some of the U α , while ensuring that X remains covered. Remark 8.3.3 -The "Hausdorff" hypothesis that I snuck in is a sanity condition which is not worth worrying about unless you're working on the algebraic geometry chapters, since all the spaces you will deal with are Hausdorff. (In fact, some authors don't even bother to include it.) For example all metric spaces are Hausdorff and thus this condition can be safely ignored if you are working with metric spaces.

What does this mean? Here's an example: But this is way overkill -we only need about four of these circles to cover the whole square. That's what is meant by a "finite subcover".

Why do we care? Because of this: Theorem 8.

3.5 (Sequentially compact ⇐⇒ compact)

A metric space M is sequentially compact if and only if it is compact.

We defer the proof to the last section.

This gives us the motivation we wanted for our definition. Sequential compactness was a condition that made sense. The open-cover definition looked strange, but it turned out to be equivalent. But we now prefer it, because we have seen that whenever possible we want to resort to open-set-only based definitions: so that e.g. they are preserved under homeomorphism.

Example 8.3.6 (An example of non-compactness)

The space X = [0, 1) is not compact in either sense. We can already see it is not sequentially compact, because it is not even complete (look at x n = 1 -1 n ). To see it is not compact under the covering definition, consider the sets

U m = 0, 1 - 1 m + 1

for m = 1, 2, . . . . Then X = U i ; hence the U i are indeed a cover. But no finite collection of the U i 's will cover X. Abuse of Notation 8.3.8. Thus, we'll never call a metric space "sequentially compact" again -we'll just say "compact". (Indeed, I kind of already did this in the previous few Let M be compact. Then M is totally bounded.

Proof using covers. For every point p ∈ M , take an ε-neighborhood of p, say U p . These cover M for the horrendously stupid reason that each point p is at the very least covered by its open neighborhood U p . Compactness then lets us take a finite subcover.

Next, an important result about maps between compact spaces.

Theorem 8.4.2 (Images of compacts are compact)

Let f : X → Y be a continuous function, where X is compact. Then the image Let X be compact and consider a continuous function f : X → R. Then f achieves a maximum value at some point, i.e. there is a point p ∈ X such that f (p) ≥ f (q) for any other q ∈ X. This function (which you plot as a hyperbola) is not bounded; essentially, you can see graphically that the issue is we can't extend it to a function on [0, 1] because it explodes near x = 0.

f img (X) ⊆ Y is compact.

One last application: if M is a compact metric space, then continuous functions f : M → N are continuous in an especially "nice" way: Definition 8.4.7. A function f : M → N of metric spaces is called uniformly continuous if for any ε > 0, there exists a δ > 0 (depending only on ε) such that whenever d M (x, y) < δ we also have d N (f (x), f (y)) < ε.

The name means that for ε > 0, we need a δ that works for every point of M . If M is compact and f : M → N is continuous, then f is uniformly continuous.

Proof using sequences. Fix ε > 0, and assume for contradiction that for every δ = 1/k there exists points x k and y k within δ of each other but with images ε > 0 apart. By compactness, take a convergent subsequence x i k → p. Then y i k → p as well, since the x k 's and y k 's are close to each other. So both sequences f (x i k ) and f (y i k ) should converge to f (p) by sequential continuity, but this can't be true since the two sequences are always ε apart.

§8.5 (Optional) Equivalence of formulations of compactness

We will prove that:

Theorem 8.5.1 (Heine-Borel for general metric spaces)

For a metric space M , the following are equivalent:

(i) Every sequence has a convergent subsequence, (ii) The space M is complete and totally bounded, and

(iii) Every open cover has a finite subcover.

We leave the proof that (i) ⇐⇒ (ii) as Problem 8F † ; the idea of the proof is much in the spirit of Theorem 8.2.2.

Proof that (i) and (ii) =⇒ (iii). We prove the following lemma, which is interesting in its own right.

Lemma 8.5.2 (Lebesgue number lemma)

Let M be a compact metric space and {U α } an open cover. Then there exists a real number δ > 0, called a Lebesgue number for that covering, such that the δ-neighborhood of any point p lies entirely in some U α .

Proof of lemma. Assume for contradiction that for every δ = 1/k there is a point x k ∈ M such that its 1/k-neighborhood isn't contained in any U α . In this way we construct a sequence x 1 , x 2 , . . . ; thus we're allowed to take a subsequence which converges to some x. Then for every ε > 0 we can find an integer n such that d(x n , x) + 1/n < ε; thus the ε-neighborhood at x isn't contained in any U α for every ε > 0. This is impossible, because we assumed x was covered by some open set. ■

Now, take a Lebesgue number δ for the covering. Since M is totally bounded, finitely many δ-neighborhoods cover the space, so finitely many U α do as well.

Proof that (iii) =⇒ (ii). Problem 8C (The cofinite topology is quasicompact only). We let X be an infinite set and equip it with the cofinite topology: the open sets are the empty set and complements of finite sets. This makes X into a topological space. Show that X is quasicompact but not Hausdorff.

Problem 8D (Cantor's intersection theorem). Let X be a compact topological space, and suppose

X = K 0 ⊇ K 1 ⊇ K 2 ⊇ . . .

is an infinite sequence of nested nonempty closed subsets. Show that n≥0 K n ̸ = ∅.

Problem 8E (Tychonoff's theorem). Let X and Y be compact metric spaces. Show that X × Y is compact. (This is also true for general topological spaces, but the proof is surprisingly hard, and we haven't even defined X × Y in general yet.)

Problem 8F † (Bolzano-Weierstraß theorem for general metric spaces). Prove that a metric space M is sequentially compact if and only if it is complete and totally bounded.

Problem 8G (Almost Arzelà-Ascoli theorem). Let f 1 , f 2 , . . . : [0, 1] → [-100, 100] be an equicontinuous sequence of functions, meaning

∀ε > 0 ∃δ > 0 ∀n ∀x, y (|x -y| < δ =⇒ |f n (x) -f n (y)| < ε)

Show that we can extract a subsequence f i 1 , f i 2 , . . . of these functions such that for every x ∈ [0, 1], the sequence 

f i 1 (x), f i 2 (x

9 Vector spaces

This is a pretty light chapter. The point of it is to define what a vector space and a basis are. These are intuitive concepts that you may already know. §9.1 The definitions of a ring and field Prototypical example for this section: Z, R, and C are rings; the latter two are fields.

I'll very informally define a ring/field here, in case you skipped the earlier chapter.

• A ring is a structure with a commutative addition and multiplication, as well as subtraction, like Z. It also has an additive identity 0 and multiplicative identity 1.

• If the multiplication is invertible like in R or C, (meaning 1 x makes sense for any x ̸ = 0), then the ring is called a field.

In fact, if you replace "field" by "R" everywhere in what follows, you probably won't lose much. It's customary to use the letter R for rings, and k or K for fields.

Finally, in case you skipped the chapter on groups, I should also mention:

• An additive abelian group is a structure with a commutative addition, as well as subtraction, plus an additive identity 0. It doesn't have to have multiplication.

A good example is R 3 (with addition componentwise).

§9.2 Modules and vector spaces

Prototypical example for this section: Polynomials of degree at most n.

You intuitively know already that R n is a "vector space": its elements can be added together, and there's some scaling by real numbers. Let's develop this more generally.

Fix a commutative ring R. My favorite example of an R-vector space is the set of polynomials of degree at most two, namely

ax 2 + bx + c | a, b, c ∈ R .

Indeed, you can add any two quadratics, and multiply by constants. You can't multiply two quadratics to get a quadratic, but that's irrelevant -in a vector space there need not be a notion of multiplying two vectors together.

In a sense we'll define later, this vector space has dimension 3 (as expected!).

Example 9.2.2 (All polynomials)

The set of all polynomials with real coefficients is an R-vector space, because you can add any two polynomials and scale by constants. (c) The set of all functions f : R → R is also a real vector space (since the notions f + g and c • f both make sense for c ∈ R).

Now let me write the actual rules for how this multiplication behaves.

Definition 9.2.5. Let R be a commutative ring. An R-module starts with an additive abelian group M = (M, +) whose identity is denoted 0 = 0 M . We additionally specify a left multiplication by elements of R. This multiplication must satisfy the following properties for r, r 1 , r 2 ∈ R and m, m 1 , m 2 ∈ M :

(i) r 1 • (r 2 • m) = (r 1 r 2 ) • m. (ii) Multiplication is distributive, meaning (r 1 + r 2 ) • m = r 1 • m + r 2 • m and r • (m 1 + m 2 ) = r • m 1 + r • m 2 . (iii) 1 R • m = m. (iv) 0 R • m = 0 M .

(This is actually extraneous; one can deduce it from the first three.)

If R is a field we say M is an R-vector space; its elements are called vectors and the members of R are called scalars.

Abuse of Notation 9.2.6. In the above, we're using the same symbol + for the addition of M and the addition of R. Sorry about that, but it's kind of hard to avoid, and the point of the axioms is that these additions should be related. I'll try to remember to put r • m for the multiplication of the module and r 1 r 2 for the multiplication of R.

Question 9.2.7. In Example 9.2.1, I was careful to say "degree at most 2" instead of "degree 2". What's the reason for this? In other words, why is

ax 2 + bx + c | a, b, c ∈ R, a ̸ = 0

not an R-vector space?

A couple less intuitive but somewhat important examples. 

ax 2 + bx + c | a, b, c ∈ R

is also a Z-module! Indeed, we can add any two such polynomials, and we can scale them by integers.

(b) The set of integers modulo 100, say Z/100Z, is a Z-module as well. Can you see how?

(c) In fact, any abelian group G = (G, +) is a Z-module. The multiplication can be defined by

n • g = g + • • • + g n times (-n) • g = n • (-g)

for n ≥ 0. (Here -g is the additive inverse of g.)

Example 9.2.9 (Every ring is its own module) (a) R can be thought of as an R-vector space over itself. Can you see why?

(b) By the same reasoning, we see that any commutative ring R can be thought of as an R-module over itself.

§9.3 Direct sums

Prototypical example for this section:

{ax 2 + bx + c} = R ⊕ xR ⊕ x 2 R

, and R 3 is the sum of its axes.

Let's return to Example 9.2.1, and consider

V = ax 2 + bx + c | a, b, c ∈ R .

Even though I haven't told you what a dimension is, you can probably see that this vector space "should have" dimension 3. We'll get to that in a moment.

The other thing you may have noticed is that somehow the x 2 , x and 1 terms don't "talk to each other". They're totally unrelated. In other words, we can consider the three sets

x 2 R := ax 2 | a ∈ R xR := {bx | b ∈ R} R := {c | c ∈ R} .

In an obvious way, each of these can be thought of as a "copy" of R.

Then V quite literally consists of the "sums of these sets". Specifically, every element of V can be written uniquely as the sum of one element from each of these sets. This motivates us to write

V = x 2 R ⊕ xR ⊕ R.

The notion which captures this formally is the direct sum. Take the vector space

R 2 = {(x, y) | x ∈ R, y ∈ R}.

We can consider it as a direct sum of its x-axis and y-axis:

X = {(x, 0) | x ∈ R} and Y = {(0, y) | y ∈ R} . Then R 2 = X ⊕ Y .

This gives us a "top-down" way to break down modules into some disconnected components.

By applying this idea in reverse, we can also construct new vector spaces as follows. In a very unfortunate accident, the two names and notations for technically distinct things are exactly the same. Definition 9.3.3. Let M and N be R-modules. We define the direct sum M ⊕ N to be the R-module whose elements are pairs (m, n) ∈ M × N . The operations are given by

(m 1 , n 1 ) + (m 2 , n 2 ) = (m 1 + m 2 , n 1 + n 2 ). and r • (m, n) = (r • m, r • n).

For example, while we technically wrote R 2 = X ⊕ Y , since each of X and Y is a copy of R, we might as well have written

R 2 ∼ = R ⊕ R.

Abuse of Notation 9.3.4. The above illustrates an abuse of notation in the way we write a direct sum. The symbol ⊕ has two meanings.

• If V is a given space and W 1 and W 2 are subspaces, then V = W 1 ⊕ W 2 means that "V splits as a direct sum W 1 ⊕ W 2 " in the way we defined above.

• If W 1 and W 2 are two unrelated spaces, then W 1 ⊕ W 2 is defined as the vector space whose elements are pairs (w

1 , w 2 ) ∈ W 1 × W 2 .

You can see that these definitions "kind of" coincide.

In this way, you can see that V should be isomorphic to R ⊕ R ⊕ R; we had V = x 2 R ⊕ xR ⊕ R, but the 1, x, x 2 don't really talk to each other and each of the summands is really just a copy of R at heart. Definition 9.3.5. We can also define, for every positive integer n, the module

M ⊕n := M ⊕ M ⊕ • • • ⊕ M n times .

§9.4 Linear independence, spans, and basis

Prototypical example for this section:

1, x, x 2 is a basis of ax 2 + bx + c | a, b, c ∈ R .

The idea of a basis, the topic of this section, gives us another way to capture the notion that Let M be an R-module now. We define three very classical notions that you likely are already familiar with. If not, fall upon your notion of Euclidean space or V above. • It is called linearly independent if there is no nontrivial linear combination with value 0 M . (Observe that 0

V = ax 2 + bx + c | a, b, c ∈ R is sums of copies of {1, x, x 2 }.

Definition 9.4.1. A linear combination of some vectors v 1 , . . . , v n is a sum of the form r 1 v 1 + • • • + r n v n , where r 1 , . . . , r n ∈ R. The linear combination is called trivial if r 1 = r 2 = • • • = r n = 0 R ,

M = 0 • v 1 + 0 • v 2 + • • • + 0 • v n

is always true -the assertion is that there is no other way to express 0 M in this form.)

• It is called a generating set if every v ∈ M can be written as a linear combination of the {v i }. If M is a vector space we say it is spanning instead.

• It is called a basis (plural bases) if every v ∈ M can be written uniquely as a linear combination of the {v i }.

The same definitions apply for an infinite set, with the proviso that all sums must be finite.

So by definition, 1, x, x 2 is a basis for V . It's not the only one: {2, x, x 2 } and {x + 4, x -2, x 2 + x} are other examples of bases, though not as natural. However, the set S = {3 + x 2 , x + 1, 5 + 2x + x 2 } is not a basis; it fails for two reasons:

• Note that 0 = (3 + x 2 ) + 2(x + 1) -(5 + 2x + x 2 ). So the set S is not linearly independent.

• It's not possible to write x 2 as a sum of elements of S. So S fails to be spanning.

With these new terms, we can say a basis is a linearly independent and spanning set. 

(a) Regard Q[ √ 2] = a + b √ 2 | a, b ∈ Q as a Q-vector space. Then {1, √ 2} is a basis.

(b) If V is the set of all real polynomials, there is an infinite basis {1, x, x 2 , . . . }. The condition that we only use finitely many terms just says that the polynomials must have finite degree (which is good).

(c) Let V = {(x, y, z) | x + y + z = 0 and x, y, z ∈ R}. Then we expect there to be a basis of size 2, but unlike previous examples there is no immediately "obvious" choice. Some working examples include:

• (1, -1, 0) and (1, 0, -1),

• (0, 1, -1) and (1, 0, -1),

• (5, 3, -8) and (2, -1, -1). Now we state a few results which assert that bases in vector spaces behave as nicely as possible.

Theorem 9.4.5 (Maximality and minimality of bases) Let V be a vector space over some field k and take e 1 , . . . , e n ∈ V . The following are equivalent: Proof. Straightforward, do it yourself if you like. The key point to notice is that you need to divide by scalars for the converse direction, hence V is required to be a vector space instead of just a module for the implications (b) =⇒ (a) and (c) =⇒ (a).

Theorem 9.4.7 (Dimension theorem for vector spaces)

If a vector space V has a finite basis, then every other basis has the same number of elements.

Proof. We prove something stronger: Assume v 1 , . Let A 0 = {v 1 , . . . , v n } be the spanning set. Throw in w 1 : by the spanning condition,

w 1 = c 1 v 1 + • • • + c n v n .

There's some nonzero coefficient, say c n . Thus

v n = 1 c n w 1 - c 1 c n v 1 - c 2 c n v 2 -. . . . Thus A 1 = {v 1 , . . . , v n-1

, w 1 } is spanning. Now do the same thing, throwing in w 2 , and deleting some element of the v i as before to get A 2 ; the condition that the w i are linearly independent ensures that some v i coefficient must always not be zero. Since we can eventually get to A m , we have n ≥ m.

Remark 9.4.9 (Generalizations)

• The theorem is true for an infinite basis as well if we interpret "the number of elements" as "cardinality". This is confusing on a first read through, so we won't elaborate.

• In fact, this is true for modules over any commutative ring. Interestingly, the proof for the general case proceeds by reducing to the case of a vector space.

The dimension theorem, true to its name, lets us define the dimension of a vector space as the size of any finite basis, if one exists. When it does exist we say V is finite-dimensional. So for example,

V = ax 2 + bx + c | a, b, c ∈ R

has dimension three, because 1, x, x 2 is a basis. That's not the only basis: we could as well have written

a(x 2 -4x) + b(x + 2) + c | a, b, c ∈ R

and gotten the exact same vector space. But the beauty of the theorem is that no matter how we try to contrive the generating set, we always will get exactly three elements.

That's why it makes sense to say V has dimension three.

On the other hand, the set of all polynomials R[x] is infinite-dimensional (which should be intuitively clear).

A basis e 1 , . . . , e n of V is really cool because it means that to specify v ∈ V , I only have to specify a 1 , . . . , a n ∈ k, and then let v = a 1 e 1 + • • • + a n e n . You can even think of v as (a 1 , . . . , a n ). To put it another way, if V is a k-vector space we always have

V = e 1 k ⊕ e 2 k ⊕ • • • ⊕ e n k.

§9.5 Linear maps

Prototypical example for this section: Evaluation of {ax 2 + bx + c} at x = 3.

We've seen homomorphisms and continuous maps. Now we're about to see linear maps, the structure preserving maps between vector spaces. Can you guess the definition? Definition 9.5.1. Let V and W be vector spaces over the same field k. A linear map is a map T : V → W such that:

(i) We have T (v 1 + v 2 ) = T (v 1 ) + T (v 2 ) for any v 1 , v 2 ∈ V . 1 (ii) For any a ∈ k and v ∈ V , T (a • v) = a • T (v).

If this map is a bijection (equivalently, if it has an inverse), it is an isomorphism. We then say V and W are isomorphic vector spaces and write V ∼ = W .

Example 9.5.2 (Examples of linear maps) (a) For any vector spaces V and W there is a trivial linear map sending everything to 0 W ∈ W .

(b) For any vector space V , there is the identity isomorphism id :

V → V . (c) The map R 3 → R by (a, b, c) → 4a + 2b + c is a linear map.

(d) Let V be the set of real polynomials of degree at most 2. The map R 3 → V by (a, b, c) → ax 2 + bx + c is an isomorphism.

(e) Let V be the set of real polynomials of degree at most 2. The map V → R by ax 2 + bx + c → 9a + 3b + c is a linear map, which can be described as "evaluation at 3".

(f) Let W be the set of functions R → R. The evaluation map W → R by f → f (0) is a linear map. (g) There is a map of Q-vector spaces Q[ √ 2] → Q[ √ 2] called "multiply by √ 2"; this map sends a + b √ 2 → 2b + a √ 2.

This map is an isomorphism, because it has an inverse "multiply by 1/ √ 2".

In the expression T (a • v) = a • T (v), note that the first • is the multiplication of V and the second • is the multiplication of W . Note that this notion of isomorphism really only cares about the size of the basis: Proposition 9.5.3 (n-dimensional vector spaces are isomorphic) If V is an n-dimensional vector space, then V ∼ = k ⊕n . Question 9.5.4. Let e 1 , . . . , e n be a basis for V . What is the isomorphism? (Your first guess is probably right.) Remark 9.5.5 -You could technically say that all finite-dimensional vector spaces are just k ⊕n and that no other space is worth caring about. But this seems kind of rude. Spaces often are more than just triples: ax 2 + bx + c is a polynomial, and so it has some "essence" to it that you'd lose if you compressed it into (a, b, c).

Moreover, a lot of spaces, like the set of vectors (x, y, z) with x + y + z = 0, do not have an obvious choice of basis. Thus to cast such a space into k ⊕n would require you to make arbitrary decisions. §9.6 What is a matrix? Now I get to tell you what a matrix is: it's a way of writing a linear map in terms of bases.

Suppose we have a finite-dimensional vector space V with basis e 1 , . . . , e m and a vector space W with basis w 1 , . . . , w 

T (a 1 e 1 + • • • + a m e m ) = a 1 T (e 1 ) + • • • + a m T (e m ).

Since the e i are a basis, that tells you all you need to know about T .

Example 9.6.1 (Extending linear maps) 

Let V = ax 2 + bx + c | a, b, c ∈ R . Then T (ax 2 + bx + c) = aT (x 2 ) + bT (x) + cT (1

T (e 1 ) = a 11 w 1 + a 21 w 2 + • • • + a n1 w n .

So rather than telling you the value of T (e 1 ) in some abstract space W , I could just tell you what a 11 , a 21 , . . . , a n1 were. Then I'd repeat this for T (e 2 ), T (e 3 ), all the way up to T (e m ), and that would tell you everything you need to know about T .

That's where the matrix T comes from! It's a concise way of writing down all mn numbers I need to tell you. To be explicit, the matrix for T is defined as the array 

T =    | | | T (e 1 ) T (e 2 ) . . . T (e m ) | | |    m columns n rows =      

T (e 1 ) = 4w 1 + 7w 2 T (e 2 ) = 2w 1 + 3w 2 T (e 3 ) = w 1

The columns then correspond to T (e 1 ), T (e 2 ), T (e 3 ):

T = 4 2 1 7 3 0

Example 9.6.3 (An example of a matrix after choosing a basis)

We again let V = ax 2 + bx + c be the vector space of polynomials of degree at most 2. We fix the basis 1, x, x 2 for it.

Consider the "evaluation at 3" map, a map V → R. We pick 1 as the basis element of the RHS; then we can write it as a 1 × 3 matrix 1 3 9 with the columns corresponding to T (1), T (x), T (x 2 ).

From here you can actually work out for yourself what it means to multiply two matrices. Suppose we have picked a basis for three spaces U , V , W . Given maps T : U → V and S : V → W , we can consider their composition S • T , i.e.

U T -→ V S -→ W.

Matrix multiplication is defined exactly so that the matrix ST is the same thing we get from interpreting the composed function S • T as a matrix.

Exercise 9.6.4. Check this for yourself! For a concrete example let R 2 T -→ R 2 S -→ R 2 by T (e 1 ) = 2e 1 + 3e 2 and T (e 2 ) = 4e 1 + 5e 2 , S(e 1 ) = 6e 1 + 7e 2 and S(e 2 ) = 8e 1 + 9e 2 . Compute S(T (e 1 )) and S(T (e 2 )) and see how it compares to multiplying the matrices associated to S and T .

In particular, since function composition is associative, it follows that matrix multiplication is as well. To drive this point home, A matrix is the laziest possible way to specify a linear map from V to W . This means you can define concepts like the determinant or the trace of a matrix both in terms of an "intrinsic" map T : V → W and in terms of the entries of the matrix. Since the map T itself doesn't refer to any basis, the abstract definition will imply that the numerical definition doesn't depend on the choice of a basis. §9.7 Subspaces and picking convenient bases Prototypical example for this section: Any two linearly independent vectors in R 3 . Definition 9.7.1. Let M be a left R-module. A submodule N of M is a module N such that every element of N is also an element of M . If M is a vector space then N is called a subspace.

Example 9.7.2 (Kernels)

The kernel of a map T : V → W (written ker T ) is the set of v ∈ V such that T (v) = 0 W . It is a subspace of V , since it's closed under addition and scaling (why?).

Example 9.7.3 (Spans) Let V be a vector space and v 1 , . . . , v m be any vectors of V . The span of these vectors is defined as the set

{a 1 v 1 + • • • + a m v m | a 1 , . . . , a m ∈ k} .

Note that it is a subspace of V as well! Question 9.7.4. Why is 0 V an element of each of the above examples? In general, why must any subspace contain 0 V ? Subspaces behave nicely with respect to bases.

Theorem 9.7.5 (Basis completion)

Let V be an n-dimensional space, and . . , e m is a basis of V ′ , then there exist e m+1 , . . . , e n in V such that e 1 , . . . , e n is a basis of V .

V ′ a subspace of V . Then (a) V ′ is also finite-dimensional. (b) If e 1 , .

Proof. Omitted, since it is intuitive and the proof is not that enlightening. (However, we will use this result repeatedly later on, so do take the time to internalize it now.)

A very common use case is picking a convenient basis for a map T .

Theorem 9.7.6 (Picking a basis for linear maps)

Let T : V → W be a map of finite-dimensional vector spaces, with n = dim V , m = dim W . Then there exists a basis v 1 , . . . , v n of V and a basis w 1 , . . . , w m of W , as well as a nonnegative integer k, such that

T (v i ) = w i if i ≤ k 0 W if i > k.

Moreover dim ker T = n -k and dim T img (V ) = k.

Sketch of Proof.

You might like to try this one yourself before reading on: it's a repeated application of Theorem 9.7.5. Let ker T have dimension n -k. We can pick v k+1 , . . . , v n a basis of ker T . Then extend it to a basis v 1 , . . . , v n of V . The map T is injective over the span of v 1 , . . . , v k (since only 0 V is in the kernel) so its images in W are linearly independent. Setting w i = T (v i ) for each i, we get some linearly independent set in W . Then extend it again to a basis of W .

This theorem is super important, not only because of applications but also because it will give you the right picture in your head of how a linear map is supposed to look. I'll even draw a cartoon of it to make sure you remember: 

e n W f 1 f 2 . . . f k f k+1 f k+2 f k+3 . . . f m T 0 0 0 im T ker T

In particular, for T : V → W , one can write V = ker T ⊕ V ′ , so that T annihilates its kernel while sending V ′ to an isomorphic copy in W .

A corollary of this (which you should have expected anyways) is the so called ranknullity theorem, which is the analog of the first isomorphism theorem. 

§9.8 A cute application: Lagrange interpolation

Here's a cute application2 of linear algebra to a theorem from high school.

Theorem 9.8.1 (Lagrange interpolation)

Let x 1 , . . . , x n+1 be distinct real numbers and y 1 , . . . , y n+1 any real numbers. Then there exists a unique polynomial P of degree at most n such that P (x i ) = y i for every i.

When n = 1 for example, this loosely says there is a unique line joining two points.

Proof. The idea is to consider the vector space V of polynomials with degree at most n, as well as the vector space W = R n+1 . Question 9.8.2. Check that dim V = n + 1 = dim W . This is easiest to do if you pick a basis for V , but you can then immediately forget about the basis once you finish this exercise.

Then consider the linear map T : V → W given by P → (P (x 1 ), . . . , P (x n+1 )) . This is indeed a linear map because, well, T (P + Q) = T (P ) + T (Q) and T (cP ) = cT (P ). It also happens to be injective: if P ∈ ker T , then P (x 1 ) = • • • = P (x n+1 ) = 0, but deg P ≤ n and so P can only be the zero polynomial.

So T is an injective map between vector spaces of the same dimension. Thus it is actually a bijection, which is exactly what we wanted. §9.9 (Digression) Arrays of numbers are evil As I'll stress repeatedly, a matrix represents a linear map between two vector spaces. Writing it in the form of an m × n matrix is merely a very convenient way to see the map concretely. But it obfuscates the fact that this map is, well, a map, not an array of numbers.

If you took high school precalculus, you'll see everything done in terms of matrices. To any typical high school student, a matrix is an array of numbers. No one is sure what exactly these numbers represent, but they're told how to magically multiply these arrays to get more arrays. They're told that the matrix

      1 0 . . . 0 0 1 . . . 0 . . . . . . . . . . . . 0 0 . . . 1      

is an "identity matrix", because when you multiply by another matrix it doesn't change.

Then they're told that the determinant is some magical combination of these numbers formed by this weird multiplication rule. No one knows what this determinant does, other than the fact that det(AB) = det A det B, and something about areas and row operations and Cramer's rule.

Then you go into linear algebra in college, and you do more magic with these arrays of numbers. You're told that two matrices T 1 and T 2 are similar if

T 2 = ST 1 S -1

for some invertible matrix S. You're told that the trace of a matrix Tr T is the sum of the diagonal entries. Somehow this doesn't change if you look at a similar matrix, but you're not sure why. Then you define the characteristic polynomial as

p T (X) = det(XI -T ).

Somehow this also doesn't change if you take a similar matrix, but now you really don't know why. And then you have the Cayley-Hamilton theorem in all its black magic: p T (T ) is the zero map. Out of curiosity you Google the proof, and you find some ad-hoc procedure which still leaves you with no idea why it's true. This is terrible. What's so special about T 2 = ST 1 S -1 ? Only if you know that the matrices are linear maps does this make sense: T 2 is just T 1 rewritten with a different choice of basis.

I really want to push the opposite view. Linear algebra is the study of linear maps, but it is taught as the study of arrays of numbers, and no one knows what these numbers mean. And for a good reason: the numbers are meaningless. They are a highly convenient way of encoding the matrix, but they are not the main objects of study, any more than the dates of events are the main objects of study in history.

The other huge downside is that people get the impression that the only (real) vector space in existence is R ⊕n . As explained in Remark 9.5.5, while you can work this way if you're a soulless robot, it's very unnatural for humans to do so.

When I took Math 55a as a freshman at Harvard, I got the exact opposite treatment: we did all of linear algebra without writing down a single matrix. During all this time I was quite confused. What's wrong with a basis? I didn't appreciate until later that this approach was the morally correct way to treat the subject: it made it clear what was happening.

Throughout the Napkin, I've tried to strike a balance between these two approaches, using matrices when appropriate to illustrate the maps and to simplify proofs, but ultimately writing theorems and definitions in their morally correct form. I hope that this has both the advantage of giving the "right" definitions while being concrete enough to be digested. But I would like to say for the record that, if I had to pick between the high school approach and the 55a approach, I would pick 55a in a heartbeat. §9.10 A word on general modules Prototypical example for this section:

Z[ √ 2] is a Z-module of rank two.

I focused mostly on vector spaces (aka modules over a field) in this chapter for simplicity, so I want to make a few remarks about modules over a general commutative ring R before concluding.

Firstly, recall that for general modules, we say "generating set" instead of "spanning set". Shrug.

The main issue with rings is that our key theorem Theorem 9.4.5 fails in spectacular ways. For example, consider Z as a Z-module over itself. Then {2} is linearly independent, but it cannot be extended to a basis. Similarly, {2, 3} is spanning, but one cannot cut it down to a basis. You can see why defining dimension is going to be difficult.

Nonetheless, there are still analogs of some of the definitions above.

Definition 9.10. 

The Z-module Z[ √ 2] = a + b √ 2 | a, b ∈ Z has a basis {1, √, so we say it is a free Z-module of rank 2.

Abuse of Notation 9.10.4 (Notation for groups). Recall that an abelian group can be viewed a Z-module (and in fact vice-versa!), so we can (and will) apply these words to abelian groups. We'll use the notation G ⊕ H for two abelian groups G and H for their Cartesian product, emphasizing the fact that G and H are abelian. This will happen when we study algebraic number theory and homology groups. §9.11 A few harder problems to think about General hint: Theorem 9.7.6 will be your best friend for many of these problems.

Problem 9A † . Let V and W be finite-dimensional vector spaces with nonzero dimension, and consider linear maps T : V → W . Complete the following table by writing "sometimes", "always", or "never" for each entry.

T injective T surjective T isomorphism If dim V > dim W . . . If dim V = dim W . . . If dim V < dim W . . .

Problem 9B † (Equal dimension vector spaces are usually isomorphisms). Let V and W be finite-dimensional vector spaces with dim V = dim W . Prove that for a map T : V → W , the following are equivalent: Problem 9G ⋆ (Idempotents are projection maps). Let P : V → V be a linear map, where V is a vector space (not necessarily finite-dimensional). Suppose P is idempotent, meaning P (P (v)) = P (v) for each v ∈ V , or equivalently P is the identity on its image.

• T is injective, • T is surjective, • T is bijective. Problem 9C (Multiplication by √ 5). Let V = Q[ √ 5] = a + b √ 5 be a two-dimensional Q-vector

Prove that V = ker P ⊕ im P.

Thus we can think of P as projection onto the subspace im P .

Problem 9H ⋆ . Let V be a finite dimensional vector space. Let T : V → V be a linear map, and let T n : V → V denote T applied n times. Prove that there exists an integer N such that V = ker T N ⊕ im T N .

10 Eigen-things

This chapter will develop the theory of eigenvalues and eigenvectors, the so-called "Jordan canonical form". (Later on we will use it to define the characteristic polynomial.) §10.1 Why you should care

We know that a square matrix T is really just a linear map from V to V . What's the simplest type of linear map? It would just be multiplication by some scalar λ, which would have associated matrix (in any basis!) Let V be some two-dimensional vector space with e 1 and e 2 as basis elements. Let's consider a map T : V → V by e 1 → 2e 1 and e 2 → e 1 + 3e 2 , which you can even write concretely as

T =       λ 0 . . . 0 0 λ .

T = 2 1 0 3 in basis e 1 , e 2 .

This doesn't look anywhere as nice until we realize we can rewrite it as

e 1 → 2e 1 e 1 + e 2 → 3(e 1 + e 2 ).

So suppose we change to the basis e 1 and e 1 + e 2 . Thus in the new basis,

T = 2 0 0 3 in basis e 1 , e 1 + e 2 .

So our completely random-looking map, under a suitable change of basis, looks like the very nice maps we described before!

In this chapter, we will be making our luck, and we will see that our better understanding of matrices gives us the right way to think about this.

§10.2 Warning on assumptions

Most theorems in this chapter only work for • finite-dimensional vector spaces V ,

• over a field k which is algebraically closed.

On the other hand, the definitions work fine without these assumptions.

§10.3 Eigenvectors and eigenvalues

Let k be a field and V a vector space over it. In the above example, we saw that there were two very nice vectors, e 1 and e 1 + e 2 , for which V did something very simple.

Naturally, these vectors have a name.

Definition 10.3.1. Let T : V → V and v ∈ V a nonzero vector. We say that v is an eigenvector if T (v) = λv for some λ ∈ k (possibly zero, but remember v ̸ = 0). The value λ is called an eigenvalue of T . We will sometimes abbreviate "v is an eigenvector with eigenvalue λ" to just "v is a λ-eigenvector".

Of course, no mention to a basis anywhere. (b) Of course, 5e 1 is also an 2-eigenvector.

(c) And, 7e 1 + 7e 2 is also a 3-eigenvector.

So you can quickly see the following observation. This lets us state succinctly that "2 is an eigenvalue of T with one-dimensional eigenspace spanned by e 1 ". Unfortunately, it's not exactly true that eigenvalues always exist.

Example 10.3.5 (Eigenvalues need not exist)

Let V = R 2 and let T be the map which rotates a vector by 90 • around the origin.

Then T (v) is not a multiple of v for any v ∈ V , other than the trivial v = 0.

However, it is true if we replace k with an algebraically closed field1 .

Theorem 10.3.6 (Eigenvalues always exist over algebraically closed fields)

Suppose k is an algebraically closed field. Let V be a finite dimensional k-vector space. Then if T : V → V is a linear map, there exists an eigenvalue λ ∈ k.

Proof. (From [Ax97] ) The idea behind this proof is to consider "polynomials" in T . For example, 2T 2 -4T + 5 would be shorthand for 2T (T (v)) -4T (v) + 5v. In this way we can consider "polynomials" P (T ); this lets us tie in the "algebraically closed" condition. These polynomials behave nicely:

Question 10.3.7. Show that P (T ) + Q(T ) = (P + Q)(T ) and P (T ) • Q(T ) = (P • Q)(T ).

Let n = dim V < ∞ and fix any nonzero vector v ∈ V , and consider vectors v, T (v), . . . , T n (v). There are n + 1 of them, so they can't be linearly independent for dimension reasons; thus there is a nonzero polynomial P such that P (T ) is zero when applied to v. WLOG suppose P is a monic polynomial, and thus P (z) = (z -r 1 ) . . . (z -r m ) say. Then we get

0 = (T -r 1 id) • (T -r 2 id) • • • • • (T -r m id)(v)

(where id is the identity matrix). This means at least one of T -r i id is not injective, i.e. has a nontrivial kernel, which is the same as an eigenvector.

So in general we like to consider algebraically closed fields. This is not a big loss: any real matrix can be interpreted as a complex matrix whose entries just happen to be real, for example. §10.4 The Jordan form So that you know exactly where I'm going, here's the main theorem.

Definition 10.4.1. A Jordan block is an n × n matrix of the following shape: 

            λ 1 0 0 . . . 0 0 0 λ 1 0 . . .

            .

In other words, it has λ on the diagonal, and 1 above it. We allow n = 1, so λ is a Jordan block.

Theorem 10.4.2 (Jordan canonical form)

Let T : V → V be a linear map of finite-dimensional vector spaces over an algebraically closed field k. Then we can choose a basis of V such that the matrix T is "block-diagonal" with each block being a Jordan block. Such a matrix is said to be in Jordan form. This form is unique up to rearranging the order of the blocks.

As an example, this means the matrix should look something like:

                 λ 1 1 0 λ 1 λ 2 λ 3 1 0 0 λ 3 1 0 0 λ 3 . . . λ m 1 0 λ m                 

Question 10.4.3. Check that diagonal matrices are the special case when each block is

1 × 1.

What does this mean? Basically, it means our dream is almost true. What happens is that V can get broken down as a direct sum

V = J 1 ⊕ J 2 ⊕ • • • ⊕ J m

and T acts on each of these subspaces independently. These subspaces correspond to the blocks in the matrix above. In the simplest case, dim J i = 1, so J i has a basis element e for which T (e) = λ i e; in other words, we just have a simple eigenvalue. But on occasion, the situation is not quite so simple, and we have a block of size greater than 1; this leads to 1's just above the diagonals.

I'll explain later how to interpret the 1's, when I make up the word descending staircase. For now, you should note that even if dim J i ≥ 2, we still have a basis element which is an eigenvector with eigenvalue λ i .

Example 10.4.4 (A concrete example of Jordan form)

Let T : k 6 → k 6 and suppose T is given by the matrix

T =         

5 0 0 0 0 0 0 2 1 0 0 0 0 0 2 0 0 0 0 0 0 7 0 0 0 0 0 0 3 0 0 0 0 0 0 3

         .

Reading the matrix, we can compute all the eigenvectors and eigenvalues: for any constants a, b ∈ k we have

T (a • e 1 ) = 5a • e 1 T (a • e 2 ) = 2a • e 2 T (a • e 4 ) = 7a • e 4 T (a • e 5 + b • e 6 ) = 3 [a • e 5 + b • e 6 ] .

The element e 3 on the other hand, is not an eigenvector since T (e 3 ) = e 2 + 2e 3 . §10.5 Nilpotent maps Bear with me for a moment. First, define:

Definition 10.5.1. A map T : V → V is nilpotent if T m

is the zero map for some integer m. (Here T m means "T applied m times".) What's an example of a nilpotent map? Example 10.5.2 (The "descending staircase") Let V = k ⊕3 have basis e 1 , e 2 , e 3 . Then the map T which sends e 3 → e 2 → e 1 → 0 is nilpotent, since T (e 1 ) = T 2 (e 2 ) = T 3 (e 3 ) = 0, and hence T 3 (v) = 0 for all v ∈ V .

The 3 × 3 descending staircase has matrix representation

T =    0 1 0 0 0 1 0 0 0    .

You'll notice this is a Jordan block.

Exercise 10.5.3. Show that the descending staircase above has 0 as its only eigenvalue.

That's a pretty nice example. As another example, we can have multiple such staircases. Picture, with some zeros omitted for emphasis:

T =        0 1 0 0 0 1 0 0 0 0 1 0 0       

You can see this isn't really that different from the previous example; it's just the same idea repeated multiple times. And in fact we now claim that all nilpotent maps have essentially that form.

Theorem 10.5.5 (Nilpotent Jordan)

Let V be a finite-dimensional vector space over an algebraically closed field k. Let T : V → V be a nilpotent map. Then we can write V = m i=1 V i where each V i has a basis of the form v i , T (v i ), . . . , T dim V i -1 (v i ) for some v i ∈ V i .

Hence:

Every nilpotent map can be viewed as independent staircases.

Each chain v i , T (v i ), T (T (v i )), .

. . is just one staircase. The proof is given later, but first let me point out where this is going.

Here's the punch line. Let's take the double staircase again. Expressing it as a matrix gives, say

S =        0 1 0 0 0 1 0 0 0 0 1 0 0        .

Then we can compute

S + λid =        λ 1 0 0 λ 1 0 0 λ λ 1 0 λ        .

It's a bunch of λ Jordan blocks! This gives us a plan to proceed: we need to break V into a bunch of subspaces such that T -λid is nilpotent over each subspace. Then Nilpotent Jordan will finish the job. §10.6 Reducing to the nilpotent case Definition 10.6.1.

Let T : V → V . A subspace W ⊆ V is called T -invariant if T (w) ∈ W for any w ∈ W .

In this way, T can be thought of as a map W → W .

In this way, the Jordan form is a decomposition of V into invariant subspaces. Now I'm going to be cheap, and define:

Definition 10.6.2. A map T : V → V is called indecomposable if it's impossible to write V = W 1 ⊕ W 2 where both W 1 and W 2 are nontrivial T -invariant spaces.

Picture of a decomposable map:

       W 1 0 0 0 0 0 0 0 0 W 2 0 0 0 0       

As you might expect, we can break a space apart into "indecomposable" parts.

Proposition 10. 6 

.3 (Invariant subspace decomposition)

Let V be a finite-dimensional vector space. Given any map T : V → V , we can write

V = V 1 ⊕ V 2 ⊕ • • • ⊕ V m

where each V i is T -invariant, and for any i the map T : V i → V i is indecomposable.

Proof. Same as the proof that every integer is the product of primes. If V is not decomposable, we are done. Otherwise, by definition write V = W 1 ⊕ W 2 and then repeat on each of W 1 and W 2 .

Incredibly, with just that we're almost done! Consider a decomposition as above, so that T :

V 1 → V 1 is an indecomposable map. Then T has an eigenvalue λ 1 , so let S = T -λ 1 id; hence ker S ̸ = {0}. Question 10.6.4. Show that V 1 is also S-invariant, so we can consider S : V 1 → V 1 .

By Problem 9H ⋆ , we have

V 1 = ker S N ⊕ im S N

for some N . But we assumed T was indecomposable, so this can only happen if im S N = {0} and ker S N = V 1 (since ker S N contains our eigenvector). Hence S is nilpotent, so it's a collection of staircases. In fact, since T is indecomposable, there is only one staircase. Hence V 1 is a Jordan block, as desired.

§10.7 (Optional) Proof of nilpotent Jordan

The proof is just induction on dim V . Assume dim V ≥ 1, and let W = T img (V ) be the image of V . Since T is nilpotent, we must have W ⊊ V . Moreover, if W = {0} (i.e. T is the zero map) then we're already done. So assume {0} ⊊ W ⊊ V .

By the inductive hypothesis, we can select a good basis of W :

B ′ = T (v 1 ), T (T (v 1 )), T (T (T (v 1 ))), . . . T (v 2 ), T (T (v 2 )), T (T (T (v 2 ))), . . . . . . , T (v ℓ ), T (T (v ℓ )), T (T (T (v ℓ ))), . . .

for some T (v i ) ∈ W (here we have taken advantage of the fact that each element of W is itself of the form T (v) for some v). Also, note that there are exactly ℓ elements of B ′ which are in ker T (namely the last element of each of the ℓ staircases). We can thus complete it to a basis v ℓ+1 , . . . , v m (where m = dim ker T ). (In other words, the last element of each staircase plus the m -ℓ new ones are a basis for ker T .) Now consider

B = v 1 , T (v 1 ), T (T (v 1 )), T (T (T (v 1 ))), . . . v 2 , T (v 2 ), T (T (v 2 )), T (T (T (v 2 ))), . . . . . . , v ℓ , T (v ℓ ), T (T (v ℓ )), T (T (T (v ℓ ))), . . . v ℓ+1 , v ℓ+2 , . . . , v m .

Question 10.7. 1 . Check that there are exactly ℓ

+ dim W + (dim ker T -ℓ) = dim V elements.

Exercise 10.7.2. Show that all the elements are linearly independent. (Assume for contradiction there is some linear dependence, then take T of both sides.)

Hence B is a basis of the desired form. §10.8 Algebraic and geometric multiplicity Prototypical example for this section: The matrix T below. This is some convenient notation: let's consider the matrix in Jordan form

T =          7 1 0 7 9 7 1 0 0 7 1 0 0 7         

.

We focus on the eigenvalue 7, which appears multiple times, so it is certainly "repeated". However, there are two different senses in which you could say it is repeated.

• Algebraic: You could say it is repeated five times, because it appears five times on the diagonal.

• Geometric: You could say it really only appears two times: because there are only two eigenvectors with eigenvalue 7, namely e 1 and e 4 .

Indeed, the vector e 2 for example has T (e 2 ) = 7e 2 + e 1 , so it's not really an eigenvector! If you apply T -7id to e 2 twice though, you do get zero. Both these notions are valid, so we will name both. To preserve generality, we first state the "intrinsic" definition.

Definition 10.8.2. Let T : V → V be a linear map and λ a scalar.

• The geometric multiplicity of λ is the dimension dim V λ of the λ-eigenspace.

• Define the generalized eigenspace V λ to be the subspace of V for which (Tλid) n (v) = 0 for some n ≥ 1. The algebraic multiplicity of λ is the dimension dim V λ .

(Silly edge case: we allow "multiplicity zero" if λ is not an eigenvalue at all.)

However in practice you should just count the Jordan blocks.

Example 10.8.3 (An example of eigenspaces via Jordan form)

Retain the matrix T mentioned earlier and let λ = 7.

• The eigenspace V λ has basis e 1 and e 4 , so the geometric multiplicity is 2.

• The generalized eigenspace V λ has basis e 1 , e 2 , e 4 , e 5 , e 6 so the algebraic multiplicity is 5.

To be completely explicit, here is how you think of these in practice:

Proposition 10.8.4 (Geometric and algebraic multiplicity vs Jordan blocks)

Assume T : V → V is a linear map of finite-dimensional vector spaces, written in Jordan form. Let λ be a scalar. Then

• The geometric multiplicity of λ is the number of Jordan blocks with eigenvalue λ; the eigenspace has one basis element per Jordan block.

• The algebraic multiplicity of λ is the sum of the dimensions of the Jordan blocks with eigenvalue λ; the eigenspace is the direct sum of the subspaces corresponding to those blocks.

Question 10.8.5. Show that the geometric multiplicity is always less than or equal to the algebraic multiplicity.

This actually gives us a tentative definition:

• The trace is the sum of the eigenvalues, counted with algebraic multiplicity.

• The determinant is the product of the eigenvalues, counted with algebraic multiplicity.

This definition is okay, but it has the disadvantage of requiring the ground field to be algebraically closed. It is also not the definition that is easiest to work with computationally.

The next two chapters will give us a better definition. §10.9 A few harder problems to think about Problem 10A (Sum of algebraic multiplicities). Given a 2018-dimensional complex vector space V and a map T : V → V , what is the sum of the algebraic multiplicities of all eigenvalues of T ?

Problem 10B (The word "diagonalizable"). A linear map T : V → V (where dim V is finite) is said to be diagonalizable if it has a basis e 1 , . . . , e n such that each e i is an eigenvector.

(a) Explain the name "diagonalizable".

(b) Suppose we are working over an algebraically closed field. Then show that T is diagonalizable if and only if for any λ, the geometric multiplicity of λ equals the algebraic multiplicity of λ.

Problem 10C (Switcharoo). Let V be the C-vector space with basis e 1 and e 2 . The map T : V → V sends T (e 1 ) = e 2 and T (e 2 ) = e 1 . Determine the eigenspaces of T .

Problem 10D. Suppose T : C ⊕2 → C ⊕2 is a linear map of C-vector spaces such that T 2011 = id. Must T be diagonalizable?

Problem 10E (Writing a polynomial backwards). Define the complex vector space V of polynomials with degree at most 2, say

V = ax 2 + bx + c | a, b, c ∈ C . Define T : V → V by T (ax 2 + bx + c) = cx 2 + bx + a.

Determine the eigenspaces of T .

Problem 10F (Differentiation of polynomials). Let V = R[x] be the infinite-dimensional real vector space of all polynomials with real coefficients. Note that d dx : V → V is a linear map (for example it sends x 3 to 3x 2 ). Which real numbers are eigenvalues of this map?

Problem 10G (Differentiation of functions). Let V be the infinite-dimensional real vector space of all infinitely differentiable functions R → R. Note that d dx : V → V is a linear map (for example it sends cos x tosin x). Which real numbers are eigenvalues of this map? You may have learned in high school that given a matrix a c b d the trace is the sum along the diagonals a + d and the determinant is ad -bc. But we know that a matrix is somehow just encoding a linear map using a choice of basis. Why would these random formulas somehow not depend on the choice of a basis?

In this chapter, we are going to give an intrinsic definition of Tr T , where T : V → V and dim V < ∞. This will give a coordinate-free definition which will in particular imply the trace a + d doesn't change if we take a different basis.

In doing so, we will introduce two new constructions: the tensor product V ⊗ W (which is a sort of product of two spaces, with dimension dim V • dim W ) and the dual space V ∨ , which is the set of linear maps V → k (a k-vector space). Later on, when we upgrade from a vector space V to an inner product space, we will see that the dual space V ∨ gives a nice interpretation of the "transpose" of a matrix. You'll already see some of that come through here.

The trace is only defined for finite-dimensional vector spaces, so if you want you can restrict your attention to finite-dimensional vector spaces for this chapter. (On the other hand we do not need the ground field to be algebraically closed.)

The next chapter will then do the same for the determinant. We know that dim(V ⊕ W ) = dim V + dim W , even though as sets V ⊕ W looks like V × W . What if we wanted a real "product" of spaces, with multiplication of dimensions?

For example, let's pull out my favorite example of a real vector space, namely

V = ax 2 + bx + c | a, b, c ∈ R .

Here's another space, a little smaller:

W = {dy + e | d, e ∈ R} .

If we take the direct sum, then we would get some rather unnatural vector space of dimension five (whose elements can be thought of as pairs (ax 2 + bx + c, dy + e)). But suppose we want a vector space whose elements are products of polynomials in V and W ; it would contain elements like 4x 2 y + 5xy + y + 3. In particular, the basis would be x 2 y, x 2 , xy, x, y, 1 and thus have dimension six.

For this we resort to the tensor product. It does exactly this, except that the "multiplication" is done by a scary1 symbol ⊗: think of it as a "wall" that separates the elements between the two vector spaces. For example, the above example might be written as

4x 2 ⊗ y + 5x ⊗ y + 1 ⊗ y + 3 ⊗ 1.

(This should be read as (4x2 ⊗ y) + (5x ⊗ y) + . . . ; addition comes after ⊗.) Of course there should be no distinction between writing 4x 2 ⊗ y and x 2 ⊗ 4y or even 2x 2 ⊗ 2y. While we want to keep the x and y separate, the scalars should be free to float around.

Of course, there's no need to do everything in terms of just the monomials. We are free to write (x + 1) ⊗ (y + 1).

If you like, you can expand this as

x ⊗ y + 1 ⊗ y + x ⊗ 1 + 1 ⊗ 1.

Same thing. The point is that we can take any two of our polynomials and artificially "tensor" them together.

The definition of the tensor product does exactly this, and nothing else.

2

Definition 11.1.1. Let V and W be vector spaces over the same field k. The tensor product V ⊗ k W is the abelian group generated by elements of the form v ⊗ w, subject to relations

(v 1 + v 2 ) ⊗ w = v 1 ⊗ w + v 2 ⊗ w v ⊗ (w 1 + w 2 ) = v ⊗ w 1 + v ⊗ w 2 (c • v) ⊗ w = v ⊗ (c • w).

As a vector space, its action is given by c

• (v ⊗ w) = (c • v) ⊗ w = v ⊗ (c • w).

Here's another way to phrase the same idea. We define a pure tensor as an element of the form v ⊗ w for v ∈ V and w ∈ W . But we let the ⊗ wall be "permeable" in the sense that

(c • v) ⊗ w = v ⊗ (c • w) = c • (v ⊗ w)

and we let multiplication and addition distribute as we expect. Then V ⊗ W consists of sums of pure tensors.

Example 11.1.2 (Infinite-dimensional example of tensor product: two-variable polynomials) Although it's not relevant to this chapter, this definition works equally well with infinite-dimensional vector spaces. The best example might be

R[x] ⊗ R R[y] = R[x, y].

That is, the tensor product of polynomials in x with real polynomials in y turns out to just be two-variable polynomials R[x, y].

Remark 11. 1.3 (Warning on sums of pure tensors) -Remember the elements of V ⊗ k W really are sums of these pure tensors! If you liked the previous example, this fact has a nice interpretation -not every polynomial in R

[x, y] = R[x] ⊗ R R[y]

factors as a polynomial in x times a polynomial in y (i.e. as pure tensors f (x) ⊗ g(y)).

But they all can be written as sums of pure tensors x a ⊗ y b .

As the example we gave suggested, the basis of V ⊗ k W is literally the "product" of the bases of V and W . In particular, this fulfills our desire that dim(V

⊗ k W ) = dim V • dim W . Proposition 11.1.4 (Basis of V ⊗ W )

Let V and W be finite-dimensional k-vector spaces. If e 1 , . . . , e m is a basis of V and f 1 , . . . , f n is a basis of W , then the basis of V ⊗ k W is precisely e i ⊗ f j , where i = 1, . . . , m and j = 1, . . . , n.

Proof. Omitted; it's easy at least to see that this basis is spanning.

Example 11. 1 

.5 (Explicit computation)

Let V have basis e 1 , e 2 and W have basis

f 1 , f 2 . Let v = 3e 1 + 4e 2 ∈ V and w = 5f 1 + 6f 2 ∈ W . Let's write v ⊗ w in this basis for V ⊗ k W : v ⊗ w = (3e 1 + 4e 2 ) ⊗ (5f 1 + 6f 2 ) = (3e 1 ) ⊗ (5f 1 ) + (4e 2 ) ⊗ (5f 1 ) + (3e 1 ) ⊗ (6f 2 ) + (4e 2 ) ⊗ (6f 2 ) = 15(e 1 ⊗ f 1 ) + 20(e 2 ⊗ f 1 ) + 18(e 1 ⊗ f 2 ) + 24(e 2 ⊗ f 2 ).

So you can see why tensor products are a nice "product" to consider if we're really interested in V × W in a way that's more intimate than just a direct sum. Remark 11.1.8 -Much like the Cartesian product A × B of sets, you can tensor together any two vector spaces V and W over the same field k; the relationship between V and W is completely irrelevant. One can think of the ⊗ as a "wall" through which one can pass scalars in k, but otherwise keeps the elements of V and W separated. Thus, ⊗ is content-agnostic.

This also means that even if V and W have some relation to each other, the tensor product doesn't remember this. So for example v

⊗ 1 ̸ = 1 ⊗ v, just like (g, 1 G ) ̸ = (1 G , g) in the group G × G. §11.

2 Dual space

Prototypical example for this section: Rotate a column matrix by 90 degrees.

Consider the following vector space:

Example 11.2.1 (Functions from R 3 → R)

The set of real functions f (x, y, z) is an infinite-dimensional real vector space. Indeed, we can add two functions to get f + g, and we can think of functions like 2f . This is a terrifyingly large vector space, but you can do some reasonable reductions. For example, you can restrict your attention to just the linear maps from R 3 to R.

That's exactly what we're about to do. This definition might seem strange at first, but bear with me.

Definition 11.2.2. Let V be a k-vector space. Then V ∨ , the dual space of V , is defined as the vector space whose elements are linear maps from V to k.

The addition and multiplication are pointwise: it's the same notation we use when we write cf + g to mean c • f (x) + g(x). The dual space itself is less easy to think about.

Let's try to find a basis for V ∨ . First, here is a very concrete interpretation of the vector space. Suppose for example V = R 3 . We can think of elements of V as column matrices, like

v =    2 5 9    ∈ V.

Then a linear map f : V → k can be interpreted as a row matrix:

f = 3 4 5 ∈ V ∨ . Then f (v) = 3 4 5    2 5 9    = 71.

More precisely: to specify a linear map V → k, I only have to tell you where each basis element of V goes. In the above example, f sends e 1 to 3, e 2 to 4, and e 3 to 5. So f sends 2e

1 + 5e 2 + 9e 3 → 2 • 3 + 5 • 4 + 9 • 5 = 71.

Let's make all this precise.

Proposition 11.2.3 (The dual basis for V ∨ )

Let V be a finite-dimensional vector space with basis e 1 , . . . , e n . For each i consider the function e ∨ i : V → k defined by

e ∨ i (e j ) = 1 i = j 0 i ̸ = j.

In more humane terms, e ∨ i (v) gives the coefficient of e i in v. 

f (e 1 ) = (3e ∨ 1 + 4e ∨ 2 + 5e ∨ 3 )(e 1 ) = 3e ∨ 1 (e 1 ) + 4e ∨ 2 (e 1 ) + 5e ∨ 3 (e 1 ) = 3 • 1 + 4 • 0 + 5 • 0 = 3.

That's exactly what we wanted.

You might be inclined to point out that V ∼ = V ∨ at this point, with an isomorphism given by e i → e ∨ i . You might call it "rotating the column matrix by 90 • ". This statement is technically true, but for a generic vector space V without any extra information, you can just think of this as an artifact of the dim V = dim V ∨ (as any two vector spaces of equal dimension are isomorphic). Most importantly, the isomorphism given above depends on what basis you picked.

Remark 11.2.5 (Explicit example showing that the isomorphism V → V ∨ given above is unnatural) Alice or Bob are looking at the same two-dimensional real vector space

V = {(x, y, z) | x + y + z = 0} .

Also, let v example = (3, 5, -8) be an example of an arbitrary element of V for concreteness.

Suppose Alice chooses the following basis vectors for V .

e 1 = (1, 0,e 2 = (0, 1, -

Alice uses this to construct an isomorphism A : V → V ∨ as described above, and considers e ∨ 1 = A(e 1 ). The element e ∨ 1 ∈ V ∨ is a function e ∨ 1 : V → R, meaning Alice can plug any vector in V into it. As an example, for v example

e ∨ 1 (v example ) = e ∨ 1 ((3, 5, -8)) = e ∨ 1 (3e 1 + 5e 2 ) = 3.

Meanwhile, Bob chooses the different basis vectors

f 1 = (1, 0, -1) f 2 = (1, -1, 0).

This gives Bob an isomorphism B : V → V ∨ , and a corresponding f ∨ 1 = B(f 1 ). Bob can also evaluate it anywhere, e.g.

f ∨ 1 (v example ) = f ∨ 1 ((3, 5, -8)) = f ∨ 1 (8f 1 -5f 2 ) = 8.

It follows that e ∨ 1 = A ((1, 0, -1)) and f ∨ 1 = B ((1, 0, -1)) are different elements of V ∨ . In other words Alice and Bob got different isomorphisms because they picked different bases. §11.3 V ∨ ⊗ W gives matrices from V to W Goal of this section: If V and W are finite-dimensional k-vector spaces then V ∨ ⊗ W represents linear maps V → W .

Here's the intuition. If V is three-dimensional and W is five-dimensional, then we can think of the maps V → W as a 5 × 3 array of numbers. We want to think of these maps as a vector space: (since one can add or scale matrices). So it had better be a vector space with dimension 15, but just saying "k ⊕15 " is not really that satisfying (what is the basis?).

To do better, we consider the tensor product

V ∨ ⊗ W

which somehow is a product of maps out of V and the target space W . We claim that this is in fact the space we want: i.e. there is a natural bijection between elements of V ∨ ⊗ W and linear maps from V to W . First, how do we interpret an element of V ∨ ⊗ W as a map V → W ? For concreteness, suppose V has a basis e 1 , e 2 , e 3 , and W has a basis

f 1 , f 2 , f 3 , f 4 , f 5 . Consider an element of V ∨ ⊗ W , say e ∨ 1 ⊗ (f 2 + 2f 4 ) + 4e ∨ 2 ⊗ f 5 .

We want to interpret this element as a function V → W : so given a v ∈ V , we want to output an element of W . There's really only one way to do this: feed in v ∈ V into the V ∨ guys on the left. That is, take the map

v → e ∨ 1 (v) • (f 2 + 2f 4 ) + 4e ∨ 2 (v) • f 5 ∈ W.

So, there's a natural way to interpret any element We can now write down something that's more true generally.

ξ 1 ⊗ w 1 + • • • + ξ m ⊗ w m ∈ V ∨ ⊗ W as a linear map V → W .

Theorem 11.3.3 (V ∨ ⊗ W ⇐⇒ linear maps V → W )

Let V and W be finite-dimensional vector spaces. We described a map

Ψ : V ∨ ⊗ W → Hom(V, W ) by sending ξ 1 ⊗ w 1 + • • • + ξ m ⊗ w m to the linear map v → ξ 1 (v)w 1 + • • • + ξ m (v)w m .

Then Ψ is an isomorphism of vector spaces, i.e. every linear map V → W can be uniquely represented as an element of V ∨ ⊗ W in this way.

The above is perhaps a bit dense, so here is a concrete example.

Example 11.3.4 (Explicit example)

Let V = R 2 and take a basis e 1 , e 2 of V . Then define T : V → V by T = 1 2 3 4 .

Then we have

Ψ(e ∨ 1 ⊗ e 1 + 2e ∨ 2 ⊗ e 1 + 3e ∨ 1 ⊗ e 2 + 4e ∨ 2 ⊗ e 2 ) = T.

The beauty is that the Ψ definition is basis-free; thus even if we change the basis, although the above expression will look completely different, the actual element in

V ∨ ⊗ V doesn't change.

Despite this, we'll indulge ourselves in using coordinates for the proof.

Proof of Theorem 11.3.3. This looks intimidating, but it's actually not difficult. We proceed in two steps:

1. First, we check that Ψ is surjective; every linear map has at least one representation in V ∨ ⊗ W . To see this, take any T : V → W . Suppose V has basis e 1 , e 2 , e 3 and that T (e 1 ) = w 1 , T (e 2 ) = w 2 and T (e 3 ) = w 3 . Then the element

e ∨ 1 ⊗ w 1 + e ∨ 2 ⊗ w 2 + e ∨ 3 ⊗ w 3

works, as it is contrived to agree with T on the basis elements e i .

2. So it suffices to check now that dim

V ∨ ⊗ W = dim Hom(V, W ). Certainly, V ∨ ⊗ W has dimension dim V •dim W . But by viewing Hom(V, W ) as dim V •dim W matrices, we see that it too has dimension dim V • dim W .

So there is a natural isomorphism V ∨ ⊗ W ∼ = Hom(V, W ). While we did use a basis liberally in the proof that it works, this doesn't change the fact that the isomorphism is "God-given", depending only on the spirit of V and W itself and not which basis we choose to express the vector spaces in.

§11.4 The trace

We are now ready to give the definition of a trace. Recall that a square matrix T can be thought of as a map T : V → V . According to the above theorem, Hom(V, V ) ∼ = V ∨ ⊗ V so every map V → V can be thought of as an element of V ∨ ⊗ V . But we can also define an evaluation map ev : V ∨ ⊗ V → k by "collapsing" each pure tensor:

f ⊗ v → f (v). So this gives us a composed map Hom(V, V ) V ∨ ⊗ V k. ∼ = ev

This result is called the trace of a matrix T .

Example 11.4.1 (Example of a trace)

Continuing the previous example,

Tr T = e ∨ 1 (e 1 ) + 2e ∨ 2 (e 1 ) + 3e ∨ 1 (e 2 ) + 4e ∨ 2 (e 2 ) = 1 + 0 + 0 + 4 = 5.

And that is why the trace is the sum of the diagonal entries. §11.5 A few harder problems to think about Problem 11A (Trace is sum of eigenvalues). Let V be an n-dimensional vector space over an algebraically closed field k. Let T : V → V be a linear map with eigenvalues λ 1 , λ 2 , . . . , λ n (counted with algebraic multiplicity). Show that Tr

T = λ 1 + • • • + λ n .

Problem 11B † (Product of traces). Let T : V → V and S : W → W be linear maps of finite-dimensional vector spaces V and W . Define T ⊗ S : Problem 11D (Putnam 1988). Let V be an n-dimensional vector space. Let T : V → V be a linear map and suppose there exists n + 1 eigenvectors, any n of which are linearly independent. Does it follow that T is a scalar multiple of the identity?

V ⊗ W → V ⊗ W by v ⊗ w → T (v) ⊗ S(w).

The goal of this chapter is to give the basis-free definition of the determinant: that is, we're going to define det T for T : V → V without making reference to the encoding for T . This will make it obvious that the determinant of a matrix does not depend on the choice of basis, and that several properties are vacuously true (e.g. that the determinant is multiplicative).

The determinant is only defined for finite-dimensional vector spaces, so if you want you can restrict your attention to finite-dimensional vector spaces for this chapter. On the other hand we do not need the ground field to be algebraically closed. §12.1 Wedge product Prototypical example for this section: Λ 2 (R 2 ) gives parallelograms.

We're now going to define something called the wedge product. It will look at first like the tensor product V ⊗ V , but we'll have one extra relation.

For simplicity, I'll first define the wedge product Λ 2 (V ). But we will later replace 2 with any n. Definition 12.1.1. Let V be a k-vector space. The 2-wedge product Λ 2 (V ) is the abelian group generated by elements of the form v ∧ w (where v, w ∈ V ), subject to the same relations

(v 1 + v 2 ) ∧ w = v 1 ∧ w + v 2 ∧ w v ∧ (w 1 + w 2 ) = v ∧ w 1 + v ∧ w 2 (c • v) ∧ w = v ∧ (c • w)

plus two additional relations:

v ∧ v = 0 and v ∧ w = -w ∧ v.

As a vector space, its action is given by c

• (v ∧ w) = (c • v) ∧ w = v ∧ (c • w). Exercise 12.1.2. Show that the condition v ∧ w = -(w ∧ v) is actually extraneous: you can derive it from the fact that v ∧ v = 0. (Hint: expand (v + w) ∧ (v + w) = 0.)

This looks almost exactly the same as the definition for a tensor product, with two subtle differences. The first is that we only have V now, rather than V and W as with the tensor product1 Secondly, there is a new mysterious relation

v ∧ v = 0 =⇒ v ∧ w = -(w ∧ v).

What's that doing there? It seems kind of weird.

I'll give you a hint.

Example 12. 1.3 (Wedge product explicit computation)

Let V = R 2 , and let v = ae 1 + be 2 , w = ce 1 + de 2 . Now let's compute v ∧ w in Λ 2 (V ). v ∧ w = (ae 1 + be 2 ) ∧ (ce 1 + de 2 )

= ac(e 1 ∧ e 1 ) + bd(e 2 ∧ e 2 ) + ad(e 1 ∧ e 2 ) + bc(e 2 ∧ e 1 )

= ad(e 1 ∧ e 2 ) + bc(e 2 ∧ e 1 ) = (ad -bc)(e 1 ∧ e 2 ).

What is ad -bc? You might already recognize it:

• You might know that the area of the parallelogram formed by v and w is ad -bc.

• You might recognize it as the determinant of a c b d . In fact, you might even know that the determinant is meant to interpret hypervolumes.

0 v = ae 1 + be 2 w = ce 1 + de 2 v + w ad -bc

This is absolutely no coincidence. The wedge product is designed to interpret signed areas. That is, v ∧ w is meant to interpret the area of the parallelogram formed by v and w. You can see why the condition (cv) ∧ w = v ∧ (cw) would make sense now. And now of course you know why v ∧ v ought to be zero: it's an area zero parallelogram!

The miracle of wedge products is that the only additional condition we need to add to the tensor product axioms is that v ∧ v = 0. Then suddenly, the wedge will do all our work of interpreting volumes for us.

Remark 12. 1.4 (Side digression on definitions in mathematics) This "property-based" philosophy is a common trope in modern mathematics. You have some intuition about an object you wish to define, and then you write down a wishlist of properties that "should" follow. But then it turns out the properties are sufficient to work with, and so for the definition, you just define an abstract object satisfying all the properties on your wishlist. Thereafter the intuition plays no "official" role; it serves only as cheerleading motivation for the wishlist.

For wedge products, the wishlist has only the single property v ∧ v = 0.

In analog to earlier:

Proposition 12.1.5 (Basis of Λ 2 (V ))

Let V be a vector space with basis e 1 , . . . , e n . Then a basis of Λ 2 (V ) is

e i ∧ e j

where i < j. Hence Λ 2 (V ) has dimension n 2 .

Proof. Surprisingly slippery, and also omitted. (You can derive it from the corresponding theorem on tensor products.)

Now I have the courage to define a multi-dimensional wedge product. It's just the same thing with more wedges. Definition 12.1.6. Let V be a vector space and m a positive integer. The space Λ m (V ) is generated by wedges of the form

v 1 ∧ v 2 ∧ • • • ∧ v m subject to relations • • • ∧ (v 1 + v 2 ) ∧ . . . = (• • • ∧ v 1 ∧ . . . ) + (• • • ∧ v 2 ∧ . . . ) • • • ∧ (cv 1 ) ∧ v 2 ∧ . . . = • • • ∧ v 1 ∧ (cv 2 ) ∧ . . . • • • ∧ v ∧ v ∧ . . . = 0 • • • ∧ v ∧ w ∧ . . . = -(• • • ∧ w ∧ v ∧ . . . ) As a vector space c • (v 1 ∧ v 2 ∧ • • • ∧ v m ) = (cv 1 ) ∧ v 2 ∧ • • • ∧ v m = v 1 ∧ (cv 2 ) ∧ • • • ∧ v m = . . . .

This definition is pretty wordy, but in English the three conditions say • We should be able to add products like before,

• You can put constants onto any of the m components (as is directly pointed out in the "vector space" action), and

• Switching any two adjacent wedges negates the whole wedge.

So this is the natural generalization of Λ 2 (V ). You can convince yourself that any element of the form

• • • ∧ v ∧ • • • ∧ v ∧ . . .

should still be zero.

Just like e 1 ∧e 2 was a basis earlier, we can find the basis for general m and n.

Proposition 12.1.7 (Basis of the wedge product)

Let V be a vector space with basis e 1 , . . . , e n . A basis for Λ m (V ) consists of the elements

e i 1 ∧ e i 2 ∧ • • • ∧ e im where 1 ≤ i 1 < i 2 < • • • < i m ≤ n.

Hence Λ m (V ) has dimension n m .

Sketch of proof. We knew earlier that e i 1 ⊗ • • • ⊗ e im was a basis for the tensor product.

Here we have the additional property that (a) if two basis elements re-appear then the whole thing becomes zero, thus we should assume the i's are all distinct; and (b) we can shuffle around elements, and so we arbitrarily decide to put the basis elements in increasing order.

§12.2 The determinant

Prototypical example for this section: (ae 1 + be 2 ) ∧ (ce 1 + de 2 ) = (ad -bc)(e 1 ∧ e 2 ). Now we're ready to define the determinant. Suppose T : V → V is a square matrix. We claim that the map Λ m (V ) → Λ m (V ) given on wedges by

v 1 ∧ v 2 ∧ • • • ∧ v m → T (v 1 ) ∧ T (v 2 ) ∧ • • • ∧ T (v m ).

and extending linearly to all of Λ m (V ) is a well-defined linear map (Here "well-defined" means that equivalent elements of the domain get mapped to equivalent elements of the codomain. This, and linearity, both follow from T being a linear map.) We call that map Λ m (T ). In other words, T (e 1 ) = ae 1 + be 2 and T (e 2 ) = ce 1 + de 2 . Now let's consider Λ 2 (V ). It has a basis e 1 ∧ e 2 . Now Λ 2 (T ) sends it to

e 1 ∧ e 2 Λ 2 (T ) ----→ T (e 1 ) ∧ T (e 2 ) = (ae 1 + be 2 ) ∧ (ce 1 + de 2 ) = (ad -bc)(e 1 ∧ e 2 ). So Λ 2 (T ) : Λ 2 (V ) → Λ 2 (V ) is multiplication by det T = ad -bc, because it sent e 1 ∧ e 2 to (ad -bc)(e 1 ∧ e 2 ).

And that is the definition of a determinant. Once again, since we defined it in terms of Λ n (T ), this definition is totally independent of the choice of basis. In other words, the determinant can be defined based on T : V → V alone without any reference to matrices. called the Leibniz formula for determinants. American high school students will recognize it; this is (unfortunately) taught as the definition of the determinant, rather than a corollary of the better definition using wedge products.

Exercise 12.2.5. Verify that expanding the wedge product yields the Leibniz formula for n = 3.

§12.3 Characteristic polynomials, and Cayley-Hamilton

Let's connect with the theory of eigenvalues. Take a map T : V → V , where V is n-dimensional over an algebraically closed field, and suppose its eigenvalues are λ 1 , λ 2 , . . . , λ n (with repetition). Then the characteristic polynomial is given by

p T (X) = (X -λ 1 )(X -λ 2 ) . . . (X -λ n ).

Note that if we've written T in Jordan form, that is, (here each * is either 0 or 1), then we can hack together the definition

T =         λ 1 * 0 . .

p T (X) := det (X • id n -T ) = det         X -λ 1 * 0 . . . 0 0 X -λ 2 * . . . 0 0 0 X -λ 3 . . . 0 . . . . . . . . . . . . . . . 0 0 0 . . . X -λ n        

.

The latter definition is what you'll see in most linear algebra books because it lets you define the characteristic polynomial without mentioning the word "eigenvalue" (i.e. entirely in terms of arrays of numbers). I'll admit it does have the merit that it means that given any matrix, it's easy to compute the characteristic polynomial and hence compute the eigenvalues; but I still think the definition should be done in terms of eigenvalues to begin with. For instance the determinant definition obscures the following theorem, which is actually a complete triviality.

Theorem 12.3.1 (Cayley-Hamilton)

Let T : V → V be a map of finite-dimensional vector spaces over an algebraically closed field. Then for any T : V → V , the map p T (T ) is the zero map.

Here, by p T (T ) we mean that if

p T (X) = X n + c n-1 X n-1 + • • • + c 0 then p T (T ) = T n + c n-1 T n-1 + • • • + c 1 T + c 0 I

is the zero map, where T k denotes T applied k times. We saw this concept already when we proved that T had at least one nonzero eigenvector.

Example 12.3.2 (Example of Cayley-Hamilton using determinant definition)

Suppose T = 1 2 3 4 . Using the determinant definition of characteristic polynomial,

we find that p T (X) = (X -1)(X -4) -(-2)(-3) = X 2 -5X -2. Indeed, you can verify that T 2 -5T -2 = 7 10 15 22 -5 • 1 2 3 4 -2 • 1 0 0 1 = 0 0 0 0 .

If you define p T without the word eigenvalue, and adopt the evil view that matrices are arrays of numbers, then this looks like a complete miracle. (Indeed, just look at the terrible proofs on Wikipedia.) But if you use the abstract viewpoint of T as a linear map, then the theorem is almost obvious:

Proof of Cayley-Hamilton. Suppose we write V in Jordan normal form as

V = J 1 ⊕ • • • ⊕ J m

where J i has eigenvalue λ i and dimension d i . By definition,

p T (T ) = (T -λ 1 ) d 1 (T -λ 2 ) d 2 . . . (T -λ m ) dm .

By definition, (T -λ 1 ) d 1 is the zero map on J 1 . So p T (T ) is zero on J 1 . Similarly it's zero on each of the other J i 's -end of story.

Remark 12.3.3 (Tensoring up) -The Cayley-Hamilton theorem holds without the hypothesis that k is algebraically closed: because for example any real matrix can be regarded as a matrix with complex coefficients (a trick we've mentioned before). I'll briefly hint at how you can use tensor products to formalize this idea.

Let's take the space V = R 3 , with basis e 1 , e 2 , e 3 . Thus objects in V are of the form r 1 e 1 + r 2 e 2 + r 3 e 3 where r 1 , r 2 , r 3 are real numbers. We want to consider essentially the same vector space, but with complex coefficients z i rather than real coefficients r i . So here's what we do: view C as a R-vector space (with basis {1, i}, say) and consider the complexification

V C := C ⊗ R V.

Then you can check that our elements are actually of the form

z 1 ⊗ e 1 + z 2 ⊗ e 2 + z 3 ⊗ e 3 .

Here, the tensor product is over R, so we have z ⊗ re i = (zr) ⊗ e i for r ∈ R. Then V C can be thought as a three-dimensional vector space over C, with basis 1 ⊗ e i for i ∈ {1, 2, 3}. In this way, the tensor product lets us formalize the idea that we "fuse on" complex coefficients. 

If T : V → W is a map, then T C : V C → W C is just the map z ⊗ v → z ⊗ T (v)

x n1 + cx n2 x n2 . . . x nn       .

Problem 12B (Determinant is product of eigenvalues). Let V be an n-dimensional vector space over an algebraically closed field k. Let T : V → V be a linear map with eigenvalues λ 1 , λ 2 , . . . , λ n (counted with algebraic multiplicity). Show that det T = λ 1 . . . λ n .

Problem 12C (Exponential matrix). Let X be an n×n matrix with complex coefficients. We define the exponential map by

exp(X) = 1 + X + X 2 2! + X 3 3! + . . .

(take it for granted that this converges to some n × n matrix). Prove that det(exp(X)) = e Tr X .

Problem 12D (Extension to Problem 9B † ). Let T : V → V be a map of finite-dimensional vector spaces. Prove that T is an isomorphism if and only if det T ̸ = 0.

Problem 12E (Based on Sweden 2010). A herd of 1000 cows of nonzero weight is given. Prove that we can remove one cow such that the remaining 999 cows cannot be partitioned into two sets with equal sum of weights.

Problem 12F (Putnam 2015). Define S to be the set of real matrices a b c d such that a, b, c, d form an arithmetic progression in that order. Find all M ∈ S such that for some integer k > 1, M k ∈ S. Problem 12G. Let V be a finite-dimensional vector space over k and T :

V → V . Show that det(a • id V -T ) = dim V n=0 a dim V -n • (-1) n Tr (Λ n (T ))

where the trace is taken by viewing Λ n (T ) : Λ n (V ) → Λ n (V ).

It will often turn out that our vector spaces which look more like R n not only have the notion of addition, but also a notion of orthogonality and the notion of distance. All this is achieved by endowing the vector space with a so-called inner form, which you likely already know as the "dot product" for R n . Indeed, in R n you already know that • v • w = 0 if and only if v and w are perpendicular, and

• |v| 2 = v • v.

The purpose is to quickly set up this structure in full generality. Some highlights of the chapter:

• We'll see that the high school "dot product" formulation is actually very natural: it falls out from the two axioms we listed above. If you ever wondered why a i b i behaves as nicely as it does, now you'll know.

• We show how the inner form can be used to make V into a metric space, giving it more geometric structure.

• A few chapters later, we'll identify V ∼ = V ∨ in a way that wasn't possible before, and as a corollary deduce the nice result that symmetric matrices with real entries always have real eigenvalues.

Throughout this chapter, all vector spaces are over C or R, unless otherwise specified. We'll generally prefer working over C instead of R since C is algebraically closed (so, e.g. we have Jordan forms). Every real matrix can be thought of as a matrix with complex entries anyways. §13.1 The inner product Prototypical example for this section: Dot product in R n . §13. 1 

.i For real numbers: bilinear forms

First, let's define the inner form for real spaces. Rather than the notation v • w it is most customary to use ⟨v, w⟩ for general vector spaces.

Definition 13.1.1. Let V be a real vector space. A real inner form1 is a function

⟨•, •⟩ : V × V → R

which satisfies the following properties:

• The form is symmetric: for any v, w ∈ V we have ⟨v, w⟩ = ⟨w, v⟩ .

Of course, one would expect this property from a product. As we already know, one can define the inner form on R n as follows. Let e 1 = (1, 0, . . . , 0), e 2 = (0, 1, . . . , 0), . . . , e n = (0, . . . , 0, 1) be the usual basis. Then we let

⟨a 1 e 1 + • • • + a n e n , b 1 e 1 + • • • + b n e n ⟩ := a 1 b 1 + • • • + a n b n .

It's easy to see this is bilinear (symmetric and linear in both arguments). To see it is positive definite, note that if

a i = b i then the dot product is a 2 1 + • • • + a 2 n

, which is zero exactly when all a i are zero. §13. 1 

.ii For complex numbers: sesquilinear forms

The definition for a complex product space is similar, but has one difference: rather than symmetry we instead have conjugate symmetry meaning ⟨v, w⟩ = ⟨w, v⟩. Thus, while we still have linearity in the first argument, we actually have a different linearity for the second argument. To be explicit: Definition 13.1.4. Let V be a complex vector space. A complex inner product is a function

⟨•, •⟩ : V × V → C

which satisfies the following properties:

• The form has conjugate symmetry, which means that for any v, w ∈ V we have ⟨v, w⟩ = ⟨w, v⟩.

• The form is sesquilinear (the name means "one-and-a-half linear"). This means that:

-The form is linear in the first argument, so again we have ⟨x + y, v⟩ = ⟨x, v⟩ + ⟨y, v⟩ ⟨cx, v⟩ = c ⟨x, v⟩ .

Again this is often abbreviated to the single line ⟨cx + y, v⟩ = c ⟨x, v⟩ + ⟨y, v⟩ in the literature.

-However, it is now anti-linear in the second argument: for any complex number c and vectors x and y we have The dot product in C n is defined as follows: let e 1 , e 2 , . . . , e n be the standard basis.

For complex numbers w i , z i we set

⟨w 1 e 1 + • • • + w n e n , z 1 e 1 + • • • + z n e n ⟩ := w 1 z 1 + • • • + w n z n .

Question 13. 1.7 . Check that the above is in fact a complex inner form. §13. 1 

.iii Inner product space

It'll be useful to treat both types of spaces simultaneously:

Definition 13.1.8. An inner product space is either a real vector space equipped with a real inner form, or a complex vector space equipped with a complex inner form.

A linear map between inner product spaces is a map between the underlying vector spaces (we do not require any compatibility with the inner form).

Remark 13.1.9 (Why sesquilinear?) -The above example explains one reason why we want to satisfy conjugate symmetry rather than just symmetry. If we had tried to define the dot product as w i z i , then we would have lost the condition of being positive definite, because there is no guarantee that ⟨v, v⟩ = z 2 i will even be a real number at all. On the other hand, with conjugate symmetry we actually enforce ⟨v, v⟩ = ⟨v, v⟩, i.e. ⟨v, v⟩ ∈ R for every v.

Let's make this point a bit more forcefully. Suppose we tried to put a bilinear form ⟨-, -⟩, on a complex vector space V . Let e be any vector with ⟨e, e⟩ = 1 (a unit vector). Then we would instead get ⟨ie, ie⟩ = -⟨e, e⟩ = -1; this is a vector with length √ -1, which is not okay! That's why it is important that, when we have a complex inner product space, our form is sesquilinear, not bilinear. Now that we have a dot product, we can talk both about the norm and orthogonality. §13.2 Norms Prototypical example for this section: R n becomes its usual Euclidean space with the vector norm.

The inner form equips our vector space with a notion of distance, which we call the norm.

Definition 13.2.1. Let V be an inner product space. The norm of v ∈ V is defined by

∥v∥ = ⟨v, v⟩.

This definition makes sense because we assumed our form to be positive definite, so ⟨v, v⟩ is a nonnegative real number.

Example 13.2.2 (R n and C n are normed vector spaces) When V = R n or V = C n with the standard dot product norm, then the norm of v corresponds to the absolute value that we are used to.

Our goal now is to prove that

With the metric d(v, w) = ∥v -w∥, V becomes a metric space.

Question 13.2.3. Verify that d(v, w) = 0 if and only if v = w.

So we just have to establish the triangle inequality. Let's now prove something we all know and love, which will be a stepping stone later: Lemma 13.2.4 (Cauchy-Schwarz) Let V be an inner product space. For any v, w ∈ V we have |⟨v, w⟩| ≤ ∥v∥ ∥w∥ with equality if and only if v and w are linearly dependent.

Proof. The theorem is immediate if ⟨v, w⟩ = 0. It is also immediate if ∥v∥ ∥w∥ = 0, since then one of v or w is the zero vector. So henceforth we assume all these quantities are nonzero (as we need to divide by them later).

The key to the proof is to think about the equality case: we'll use the inequality ⟨cv -w, cv -w⟩ ≥ 0. Deferring the choice of c until later, we compute 0 ≤ ⟨cv -w, cv -w⟩ = ⟨cv, cv⟩ -⟨cv, w⟩ -⟨w, cv⟩ + ⟨w, w⟩ = |c| 2 ⟨v, v⟩ -c ⟨v, w⟩ -c ⟨w, v⟩ + ⟨w, w⟩ In this way, our vector space now has a topological structure of a metric space.

= |c| 2 ∥v∥ 2 + ∥w∥ 2 -c

§13.3 Orthogonality

Prototypical example for this section: Still R n ! Our next goal is to give the geometric notion of "perpendicular". The definition is easy enough: Definition 13.3.1. Two nonzero vectors v and w in an inner product space are orthogonal if ⟨v, w⟩ = 0.

As we expect from our geometric intuition in R n , this implies independence: Lemma 13.3.2 (Orthogonal vectors are independent) Any set of pairwise orthogonal vectors v 1 , v 2 , . . . , v n , with ∥v i ∥ ̸ = 0 for each i, is linearly independent.

Proof. Consider a dependence

a 1 v 1 + • • • + a n v n = 0 for a i in R or C. Then 0 = v 1 , a i v i = a 1 ∥v 1 ∥ 2 .

Hence a 1 = 0, since we assumed

∥v 1 ∥ ̸ = 0. Similarly a 2 = • • • = a m = 0.

In light of this, we can now consider a stronger condition on our bases:

Definition 13.3.3. An orthonormal basis of a finite-dimensional inner product space V is a basis e 1 , . . . , e n such that ∥e i ∥ = 1 for every i and ⟨e i , e j ⟩ = 0 for any i ̸ = j.

Example 13.3.4 (R n and C n have standard bases)

In R n and C n equipped with the standard dot product, the standard basis e 1 , . . . , e n is also orthonormal.

This is no loss of generality:

Theorem 13.3.5 (Gram-Schmidt)

Let V be a finite-dimensional inner product space. Then it has an orthonormal basis.

Sketch of Proof. One constructs the orthonormal basis explicitly from any basis e 1 , . . . , e n of V . Define proj u (v) = ⟨v,u⟩ ⟨u,u⟩ u. Then recursively define

u 1 = e 1 u 2 = e 2 -proj u 1 (e 2 )

u 3 = e 3 -proj u 1 (e 3 ) -proj u 2 (e 3 ) . . .

u n = e n -proj u 1 (e n ) -• • • -proj u n-1 (e n ).

One can show the u i are pairwise orthogonal and not zero.

Thus, we can generally assume our bases are orthonormal.

Worth remarking:

Example 13.3.6 (The dot product is the "only" inner form)

Let V be a finite-dimensional inner product space, and consider any orthonormal basis e 1 , . . . , e n . Then we have that

⟨a 1 e 1 + • • • + a n e n , b 1 e 1 + • • • + b n e n ⟩ = n i,j=1 a i b j ⟨e i , e j ⟩ = n i=1 a i b i

owing to the fact that the {e i } are orthonormal.

And now you know why the dot product expression is so ubiquitous.

§13.4 Hilbert spaces

In algebra we are usually scared of infinity, and so when we defined a basis of a vanilla vector space many chapters ago, we only allowed finite linear combinations. However, if we have an inner product space, then it is a metric space and we can sometimes actually talk about convergence.

Here is how it goes:

Definition 13.4.1. A Hilbert space is an inner product space V , such that the corresponding metric space is complete.

In that case, it will now often make sense to take infinite linear combinations, because we can look at the sequence of partial sums and let it converge. Here is how we might do it. Let's suppose we have e 1 , e 2 , . . . an infinite sequence of vectors with norm 1 and which are pairwise orthogonal. Suppose c 1 , c 2 , . . . , is a sequence of real or complex numbers. Then consider the sequence 

v 1 = c 1 e 1 v 2 = c 1 e 1 + c 2 e 2 v 3 = c 1 e 1 + c 2 e 2 + c

∥v j -v i ∥ 2 = |c i+1 | 2 + • • • + |c j | 2

tends to zero as i and j get large. This is equivalent to the sequence 

s n = |c 1 | 2 + • • • + |c n | 2 being

|c i | 2 < ∞.

Thus, when we have a Hilbert space, we change our definition slightly:

Definition 13.4.3. An orthonormal basis for a Hilbert space V is a (possibly infinite) sequence e 1 , e 2 , . . . , of vectors such that • ⟨e i , e i ⟩ = 1 for all i,

• ⟨e i , e j ⟩ = 0 for i ̸ = j, i.e. the vectors are pairwise orthogonal

• every element of V can be expressed uniquely as an infinite linear combination

i c i e i

where i |c i | 2 < ∞, as described above.

That's the official definition, anyways. (Note that if dim V < ∞, this agrees with our usual definition, since then there are only finitely many e i .) But for our purposes you can mostly not worry about it and instead think:

A Hilbert space is an inner product space whose basis requires infinite linear combinations, not just finite ones.

The technical condition |c i | 2 < ∞ is exactly the one which ensures the infinite sum makes sense. §13.5 A few harder problems to think about Problem 13A (Pythagorean theorem). Show that if ⟨v, w⟩ = 0 in an inner product space, then ∥v∥ 2 + ∥w∥ 2 = ∥v + w∥ 2 . Problem 13B ⋆ (Finite-dimensional =⇒ Hilbert). Show that a finite-dimensional inner product space is a Hilbert space. Is this a Hilbert space? If so, identify a Hilbert basis.

Problem 13G (Kuratowski embedding).

A Banach space is a normed vector space V , such that the corresponding metric space is complete. (So a Hilbert space is a special case of a Banach space.) Let (M, d) be any metric space. Prove that there exists a Banach space X and an injective function f : M → X such that d(x, y) = ∥f (x) -f (y)∥ for any x and y. Now that we've worked hard to define abstract inner product spaces, I want to give an (optional) application: how to set up Fourier analysis correctly, using this language.

For fun, I also prove a form of Arrow's Impossibility Theorem using binary Fourier analysis.

In what follows, we let T = R/Z denote the "circle group", thought of as the additive group of "real numbers modulo 1". There is a canonical map e : T → C sending T to the complex unit circle, given by e(θ) = exp(2πiθ).

§14.1 Synopsis

Suppose we have a domain Z and are interested in functions f : Z → C. Naturally, the set of such functions form a complex vector space. We like to equip the set of such functions with a positive definite inner product.

The idea of Fourier analysis is to then select an orthonormal basis for this set of functions, say (e ξ ) ξ , which we call the characters; the indexing ξ are called frequencies.

In that case, since we have a basis, every function f : Z → C becomes a sum

f (x) = ξ f (ξ)e ξ

where f (ξ) are complex coefficients of the basis; appropriately we call f the Fourier coefficients. The variable x ∈ Z is referred to as the physical variable. This is generally good because the characters are deliberately chosen to be nice "symmetric" functions, like sine or cosine waves or other periodic functions. Thus we decompose an arbitrarily complicated function into a sum of nice ones.

§14.2 A reminder on Hilbert spaces

For convenience, we record a few facts about orthonormal bases.

Proposition 14.2.1 (Facts about orthonormal bases)

Let V be a complex Hilbert space with inner form ⟨-, -⟩ and suppose x = ξ a ξ e ξ and y = ξ b ξ e ξ where e ξ are an orthonormal basis. Then

⟨x, x⟩ = ξ |a ξ | 2 a ξ = ⟨x, e ξ ⟩ ⟨x, y⟩ = ξ a ξ b ξ .

Exercise 14.2.2. Prove all of these. (You don't need any of the preceding section, it's only there to motivate the notation with lots of scary ξ's. ) In what follows, most of the examples will be of finite-dimensional inner product spaces (which are thus Hilbert spaces), but the example of "square-integrable functions" will actually be an infinite dimensional example. Fortunately, as I alluded to earlier, this is no cause for alarm and you can mostly close your eyes and not worry about infinity.

§14.3 Common examples §14.3.i Binary Fourier analysis on {±1} n

Let Z = {±1} n for some positive integer n, so we are considering functions f (x 1 , . . . , x n ) accepting binary values. Then the functions Z → C form a 2 n -dimensional vector space C Z , and we endow it with the inner form

⟨f, g⟩ = 1 2 n x∈Z f (x)g(x).

In particular,

⟨f, f ⟩ = 1 2 n x∈Z |f (x)| 2

is the average of the squares; this establishes also that ⟨-, -⟩ is positive definite.

In that case, the multilinear polynomials form a basis of C Z , that is the polynomials

χ S (x 1 , . . . , x n ) = s∈S x s .

Exercise 14.3.1. Show that they're actually orthonormal under ⟨-, -⟩. This proves they form a basis, since there are 2 n of them.

Thus our frequency set is actually the subsets S ⊆ {1, . Let n = 2. Then binary functions {±1} 2 → C have a basis given by the four polynomials 1,

x 1 , x 2 , x 1 x 2 .

For example, consider the function f which is 1 at (1, 1) and 0 elsewhere. Then we can put

f (x 1 , x 2 ) = x 1 + 1 2 • x 2 + 1 2 = 1 4 (1 + x 1 + x 2 + x 1 x 2 ) .

So the Fourier coefficients are f (S) = 1 4 for each of the four S's.

This notion is useful in particular for binary functions f : {±1} n → {±1}; for these functions (and products thereof), we always have ⟨f, f ⟩ = 1.

It is worth noting that the frequency ∅ plays a special role:

Exercise 14.3.3. Show that f (∅) = 1 |Z| x∈Z f (x).

§14.3.ii Fourier analysis on finite groups Z

This time, suppose we have a finite abelian group Z, and consider functions Z → C; this is a |Z|-dimensional vector space. The inner product is the same as before:

⟨f, g⟩ = 1 |Z| x∈Z f (x)g(x).

To proceed, we'll need to be able to multiply two elements of Z. This is a bit of a nuisance since it actually won't really matter what map I pick, so I'll move briskly; feel free to skip most or all of the remaining paragraph. Definition 14.3.4. We select a symmetric non-degenerate bilinear form

• : Z × Z → T

satisfying the following properties:

• ξ • (x 1 + x 2 ) = ξ • x 1 + ξ • x 2 and (ξ 1 + ξ 2 ) • x = ξ 1 • x + ξ 2 • x (this is the word "bilinear") • • is symmetric,

• For any ξ ̸ = 0, there is an x with ξ • x ̸ = 0 (this is the word "nondegenerate").

Example 14.3.5 (The form on Z/nZ)

If Z = Z/nZ then ξ • x = (ξx)/n satisfies the above.

In general, it turns out finite abelian groups decompose as the sum of cyclic groups (see Section 18.1), which makes it relatively easy to find such a •; but as I said the choice won't matter, so let's move on. Now for the fun part: defining the characters.

Proposition 14.3.6 (e ξ are orthonormal)

For each ξ ∈ Z we define the character

e ξ (x) = e(ξ • x).

The |Z| characters form an orthonormal basis of the space of functions Z → C.

Proof. I recommend skipping this one, but it is:

e ξ , e ξ ′ = 1 |Z| x∈Z e(ξ • x)e(ξ ′ • x) = 1 |Z| x∈Z e(ξ • x)e(-ξ ′ • x) = 1 |Z| x∈Z e (ξ -ξ ′ ) • x .

In this way, the set of frequencies is also Z, but the ξ ∈ Z play very different roles from the "physical" x ∈ Z. Here is an example which might be enlightening.

Example 14.3.7 (Cube roots of unity filter) Suppose Z = Z/3Z, with the inner form given by ξ • x = (ξx)/3. Let ω = exp( 2 3 πi) be a primitive cube root of unity. Note that

e ξ (x) =        1 ξ = 0 ω x ξ = 1 ω 2x ξ = 2. Then given f : Z → C with f (0) = a, f (1) = b, f (2) = c, we obtain f (x) = a + b + c 3 • 1 + a + ω 2 b + ωc 3 • ω x + a + ωb + ω 2 c 3 • ω 2x .

In this way we derive that the transforms are

f (0) = a + b + c 3 f (1) = a + ω 2 b + ωc 3 f (2) = a + ωb + ω 2 c 3 .

Exercise 14.3.8. Show that in analogy to f (∅) for binary Fourier analysis, we now have

f (0) = 1 |Z| x∈Z f (x).

Olympiad contestants may recognize the previous example as a "roots of unity filter", which is exactly the point. For concreteness, suppose one wants to compute

1000 0 + 1000 3 + • • • + 1000 999 .

In that case, we can consider the function

w : Z/3 → C.

such that w(0) = 1 but w(1) = w(2) = 0. By abuse of notation we will also think of w as a function w : Z ↠ Z/3 → C. Then the sum in question is

n 1000 n w(n) = n 1000 n k=0,1,2 w(k)ω kn = k=0,1,2 w(k) n 1000 n ω kn = k=0,1,2 w(k)(1 + ω k ) 1000 .

In our situation, we have w(0) = w(1) = w(2) =1 3 , and we have evaluated the desired sum. More generally, we can take any periodic weight w and use Fourier analysis in order to interchange the order of summation.

Example 14.3.9 (Binary Fourier analysis) Suppose Z = {±1} n , viewed as an abelian group under pointwise multiplication hence isomorphic to (Z/2Z) ⊕n . Assume we pick the dot product defined by

ξ • x := 1 2 i ξ i -1 2 • x i -1 2

where ξ = (ξ 1 , . . . , ξ n ) and x = (x 1 , . . . , x n ).

We claim this coincides with the first example we gave. Indeed, let S ⊆ {1, . . . , n} and let ξ ∈ {±1} n which is -1 at positions in S, and +1 at positions not in S. Then the character χ S from the previous example coincides with the character e ξ in the new notation. In particular, f (S) = f (ξ).

Thus Fourier analysis on a finite group Z subsumes binary Fourier analysis. For each integer n, define e n (x) = exp(inx).

Then e n form an orthonormal basis of the Hilbert space L 2 ([-π, π]).

Thus this time the frequency set Z is infinite, and we have

f (x) = n f (n) exp(inx) almost everywhere for coefficients f (n) with n f (n) 2 < ∞.

Since the frequency set is indexed by Z, we call this a Fourier series to reflect the fact that the index is n ∈ Z. 

§14.5 Parseval and friends

Here is a fun section in which you get to learn a lot of big names quickly. Basically, we can take each of the three results from Proposition 14.2.1, translate it into the context of our Fourier analysis (for which we have an orthonormal basis of the Hilbert space), and get a big-name result.

Corollary 14.5.1 (Parseval theorem)

Let f : Z → C, where Z is a finite abelian group. Then ξ | f (ξ)| 2 = 1 |Z| x∈Z |f (x)| 2 . Similarly, if f : [-π, π] → C is square-integrable then its Fourier series satisfies n | f (n)| 2 = 1 2π [-π,π] |f (x)| 2 dx.

Proof. Recall that ⟨f, f ⟩ is equal to the square sum of the coefficients.

Corollary 14.5.2 (Fourier inversion formula)

Let f : Z → C, where Z is a finite abelian group. Then

f (ξ) = 1 |Z| x∈Z f (x)e ξ (x).

Similarly, if f : [-π, π] → C is square-integrable then its Fourier series is given by

f (n) = 1 2π [-π,π] f (x) exp(-inx) dx.

Proof. Recall that in an orthonormal basis (e ξ ) ξ , the coefficient of e ξ in f is ⟨f, e ξ ⟩. 

Similarly, if f : [-π, π] → C is square-integrable then ⟨f, g⟩ = n f (n) g(n).

Question 14.5.5. Prove this one in one line (like before).

§14.6 Application: Basel problem

One cute application about Fourier analysis on L 2 ([-π, π]) is that you can get some otherwise hard-to-compute sums, as long as you are willing to use a little calculus.

Here is the classical one:

Theorem 14.6.1 (Basel problem)

We have

n≥1 1 n 2 = π 2 6 .

The proof is to consider the identity function f (x) = x, which is certainly square-integrable.

Then by Parseval, we have

n∈Z f (n) 2 = ⟨f, f ⟩ = 1 2π [-π,π] |f (x)| 2 dx.

A calculus computation gives

1 2π [-π,π] x 2 dx = π 2 3 .

On the other hand, we will now compute all Fourier coefficients. We have already that

f (0) = 1 2π [-π,π] f (x) dx = 1 2π [-π,π]

x dx = 0.

For n ̸ = 0, we have by definition (or "Fourier inversion formula", if you want to use big words) the formula

f (n) = ⟨f, exp(inx)⟩ = 1 2π [-π,π] x • exp(inx) dx = 1 2π [-π,π] x exp(-inx) dx.

The anti-derivative is equal to 1 n 2 exp(-inx)(1 + inx), which thus with some more calculation gives that

f (n) = (-1) n n i. So n f (n) 2 = 2 n≥1 1 n 2

implying the result.

§14.7 Application: Arrow's Impossibility Theorem

As an application of binary Fourier analysis, we now prove a form of Arrow's theorem.

Consider n voters voting among 3 candidates A, B, C. Each voter specifies a tuple v i = (x i , y i , z i ) ∈ {±1} 3 as follows:

• x i = 1 if person i ranks A ahead of B, and x i = -1 otherwise.

• y i = 1 if person i ranks B ahead of C, and y i = -1 otherwise.

• z i = 1 if person i ranks C ahead of A, and z i = -1 otherwise.

Tacitly, we only consider 3! = 6 possibilities for v i : we forbid "paradoxical" votes of the form x i = y i = z i by assuming that people's votes are consistent (meaning the preferences are transitive).

For brevity, let x • = (x 1 , . . . , x n ) and define y • and z • similarly. Then, we can consider a voting mechanism

f : {±1} n → {±1} g : {±1} n → {±1} h : {±1} n → {±1} such that • f (x • ) is the global preference of A vs. B, • g(y • ) is the global preference of B vs. C,

• and h(z • ) is the global preference of C vs. A.

We'd like to avoid situations where the global preference (f (x • ), g(y • ), h(z • )) is itself paradoxical.

Let Ef denote the average value of f across all 2 n inputs. Define Eg and Eh similarly. We'll add an assumption that Ef = Eg = Eh = 0, which provides symmetry (and e.g. excludes the possibility that f , g, h are constant functions which ignore voter input). With that we will prove the following result: Theorem 14.7.1 (Arrow Impossibility Theorem) Assume that (f, g, h) always avoids paradoxical outcomes, and assume Ef = Eg = Eh = 0. Then (f, g, h) is either a dictatorship or anti-dictatorship: there exists a "dictator" k such that

f (x • ) = ±x k , g(y • ) = ±y k , h(z • ) = ±z k

where all three signs coincide.

Unlike the usual Arrow theorem, we do not assume that f (+1, . . . , +1) = +1 (hence possibility of anti-dictatorship).

Proof. Suppose the voters each randomly select one of the 3! = 6 possible consistent votes. In Problem 14B it is shown that the exact probability of a paradoxical outcome for any functions f , g, h is given exactly by

1 4 + 1 4 S⊆{1,...,n} - 1 3 |S| f (S) g(S) + g(S) h(S) + h(S) f (S) .

Assume that this probability (of a paradoxical outcome) equals 0. Then, we derive 1 = S⊆{1,...,n} --1 3

|S| f (S) g(S) + g(S) h(S) + h(S) f (S)

.

But now we can just use weak inequalities. We have f (∅) = Ef = 0 and similarly for g and h, so we restrict attention to |S| ≥ 1. We then combine the famous inequality |ab + bc + ca| ≤ a 2 + b 2 + c 2 (which is true across all real numbers) to deduce that

1 = S⊆{1,...,n} -- 1 3 |S| f (S) g(S) + g(S) h(S) + h(S) f (S) ≤ S⊆{1,...,n} 1 3 |S| f (S) 2 + g(S) 2 + h(S) 2 ≤ S⊆{1,...,n} 1 3 1 f (S) 2 + g(S) 2 + h(S) 2 = 1 3 (1 + 1 + 1) = 1.

with the last step by Parseval. So all inequalities must be sharp, and in particular f , g, h are supported on one-element sets, i. Problem 14B. Let f, g, h : {±1} n → {±1} be any three functions. For each i, we randomly select (x i , y i , z i ) ∈ {±1} 3 subject to the constraint that not all are equal (hence, choosing among 2 3 -2 = 6 possibilities). Prove that the probability that

f (x 1 , . . . , x n ) = g(y 1 , . . . , y n ) = h(z 1 , . . . , z n )

is given by the formula

1 4 + 1 4 S⊆{1,...,n} - 1 3 |S| f (S) g(S) + g(S) h(S) + h(S) f (S)

15 Duals, adjoint, and transposes

This chapter is dedicated to the basis-free interpretation of the transpose and conjugate transpose of a matrix.

Poster corollary: we will see that symmetric matrices with real coefficients are diagonalizable and have real eigenvalues.

§15.1 Dual of a map

Prototypical example for this section: The example below.

We go ahead and now define a notion that will grow up to be the transpose of a matrix. Definition 15.1.1. Let V and W be vector spaces. Suppose T : V → W is a linear map. Then we actually get a map 

T ∨ : W ∨ → V ∨ f → f • T.

T (e 1 ) = f 1 + 2f 2 T (e 2 ) = 3f 1 + 4f 2 T (e 3 ) = 5f 1 + 6f 2 .

Now consider V ∨ with its dual basis e ∨ 1 , e ∨ 2 , e ∨ 3 and W ∨ with its dual basis

f ∨ 1 , f ∨ 2 . Let's compute T ∨ (f ∨ 1 ) = f ∨ 1 • T : it is given by f ∨ 1 (T (ae 1 + be 2 + ce 3 )) = f ∨ 1 ((a + 3b + 5c)f 1 + (2a + 4b + 6c)f 2 ) = a + 3b + 5c.

So accordingly we can write

T ∨ (f ∨ 1 ) = e ∨ 1 + 3e ∨ 2 + 5e ∨ 3 Similarly, T ∨ (f ∨ 2 ) = 2e ∨ 1 + 4e ∨ 2 + 6e ∨ 3 . This determines T ∨ completely.

If we write the matrices for T and T ∨ in terms of our basis, we now see that

T = 1 3 5 2 4 6 and T ∨ =    1 2 3 4 5 6    .

So in our selected basis, we find that the matrices are transposes: mirror images of each other over the diagonal. Of course, this should work in general.

Theorem 15. 1.3 (Transpose interpretation of T ∨ )

Let V and W be finite-dimensional k-vector spaces. Then, for any T : V → W , the following two matrices are transposes:

• The matrix for T : V → W expressed in the basis (e i ), (f j ).

• The matrix for T ∨ : W ∨ → V ∨ expressed in the basis (f ∨ j ), (e ∨ i ).

Proof. The (i, j)th entry of the matrix T corresponds to the coefficient of f i in T (e j ), which corresponds to the coefficient of e ∨ j in

f ∨ i • T .

The nice part of this is that the definition of T ∨ is basis-free. So it means that if we start with any linear map T , and then pick whichever basis we feel like, then T and T ∨ will still be transposes.

§15.2 Identifying with the dual space

For the rest of this chapter, though, we'll now bring inner products into the picture. Earlier I complained that there was no natural isomorphism V ∼ = V ∨ . But in fact, given an inner form we can actually make such an identification: that is we can naturally associate every linear map ξ : V → k with a vector v ∈ V .

To see how we might do this, suppose V = R 3 for now with an orthonormal basis e 1 , e 2 , e 3 . How might we use the inner product to represent a map from V → R? For example, take ξ ∈ V ∨ by ξ(e 1 ) = 3, ξ(e 2 ) = 4 and ξ(e 3 ) = 5. Actually, I claim that ξ(v) = ⟨v, 3e 1 + 4e 2 + 5e 3 ⟩ for every v. And this works beautifully in the real case.

Theorem 15.2.2 (V ∼ = V ∨ for real inner form)

Let V be a finite-dimensional real inner product space and V ∨ its dual. Then the map

V → V ∨ by v → ⟨-, v⟩ ∈ V ∨

is an isomorphism of real vector spaces.

Proof. It suffices to show that the map is injective and surjective.

• Injective: suppose ⟨v 1 , v⟩ = ⟨v 2 , v⟩ for every vector v ∈ V . This means ⟨v 1 -v 2 , v⟩ = 0 for every vector v ∈ V . This can only happen if v 1 -v 2 = 0; for example, take v = v 1 -v 2 and use positive definiteness.

• Surjective: take an orthonormal basis e 1 , . . . e n and let e ∨ 1 , . . . , e ∨ n be the dual basis on V ∨ . Then e 1 maps to e ∨ 1 , et cetera.

Actually, since we already know dim V = dim V ∨ we only had to prove one of the above.

As a matter of personal taste, I find the proof of injectivity more elegant, and the proof of surjectivity more enlightening, so I included both. Thus

If a real inner product space V is given an inner form, then V and V ∨ are canonically isomorphic.

Unfortunately, things go awry if V is complex. Here is the results:

Theorem 15.2.3 (V versus V ∨ for complex inner forms)

Let V be a finite-dimensional complex inner product space and V ∨ its dual. Then the map

V → V ∨ by v → ⟨-, v⟩ ∈ V ∨ is a bijection of sets.

Wait, what? Well, the proof above shows that it is both injective and surjective, but why is it not an isomorphism? The answer is that it is not a linear map: since the form is sesquilinear we have for example

iv → ⟨-, iv⟩ = -i ⟨-, v⟩

which has introduced a minus sign! In fact, it is an anti-linear map, in the sense we defined before.

Eager readers might try to fix this by defining the isomorphism v → ⟨v, -⟩ instead. However, this also fails, because the right-hand side is not even an element of V ∨ : it is an "anti-linear", not linear.

And so we are stuck. Fortunately, we will only need the "bijection" result for what follows, so we can continue on anyways. (If you want to fix this, Problem 15D gives a way to do so.) §15.3 The adjoint (conjugate transpose)

We will see that, as a result of the flipping above, the conjugate transpose is actually the better concept for inner product spaces: since it can be defined using only the inner product without making mention to dual spaces at all. Definition 15.3.1. Let V and W be finite-dimensional inner product spaces, and let T : V → W . The adjoint (or conjugate transpose) of T , denoted T † : W → V , is defined as follows: for every vector w ∈ W , we let T † (w) ∈ V be the unique vector with v, T † (w)

V = ⟨T (v), w⟩ W for every v ∈ V .

Some immediate remarks about this definition:

• Our T † is well-defined, because ⟨T (-), w⟩ W is some function in V ∨ , and hence by the bijection earlier it should be uniquely of the form ⟨-, v⟩ for some v ∈ V .

• This map T † is indeed a linear map (why?).

• The niceness of this definition is that it doesn't make reference to any basis or even V ∨ , so it is the "right" definition for a inner product space.

• By symmetry, of course, we also have T † (v), w = ⟨v, T (w)⟩.

Example 15.3.2 (Example of an adjoint map)

We'll work over C, so the conjugates are more visible. Let's consider V with orthonormal basis e 1 , e 2 , e 3 and W with orthonormal basis f 1 , f 2 . We put

T (e 1 ) = if 1 + 2f 2 T (e 2 ) = 3f 1 + 4f 2 T (e 3 ) = 5f 1 + 6if 2 .

We compute T † (f 1 ). It is the unique vector x ∈ V such that

⟨v, x⟩ V = ⟨T (v), f 1 ⟩ W for any v ∈ V .

If we expand v = ae 1 + be 2 + ce 3 the above equality becomes

⟨ae 1 + be 2 + ce 3 , x⟩ V = ⟨T (ae 1 + be 2 + ce 3 ), f 1 ⟩ W = ia + 3b + 5c.

However, since x is in the second argument, this means we actually want to take T † (f 1 ) = -ie 1 + 3e 2 + 5e 3 so that the sesquilinearity will conjugate the i.

The pattern continues, though we remind the reader that we need the basis to be orthonormal to proceed.

Theorem 15.3.3 (Adjoints are conjugate transposes)

Fix an orthonormal basis of a finite-dimensional inner product space V . Let T : V → V be a linear map. If we write T as a matrix in this basis, then the matrix T † (in the same basis) is the conjugate transpose of the matrix of T ; that is, the (i, j)th entry of T ∨ is the complex conjugate of the (j, i)th entry of T .

Proof. One-line version: take v and w to be basis elements, and this falls right out. in this basis e 1 , . . . , e n . Then, letting w = e i and v = e j we deduce that e i , T † (e j ) = ⟨T (e i ), e j ⟩ = a ji =⇒ T † (e j ), e i = a ji for any i, which is enough to deduce the result.

§15.4 Eigenvalues of normal maps

We now come to the advertised theorem. Restrict to the situation where T : V → V . You see, the world would be a very beautiful place if it turned out that we could pick a basis of eigenvectors that was also orthonormal. This is of course far too much to hope for; even without the orthonormal condition, we saw that Jordan form could still have 1's off the diagonal. However, it turns out that there is a complete characterization of exactly when our overzealous dream is true.

Definition 15.4.1. We say a linear map T (from a finite-dimensional inner product space to itself) is normal if T T † = T † T .

We say a complex T is self-adjoint or Hermitian if T = T † ; i.e. as a matrix in any orthonormal basis, T is its own conjugate transpose. For real T we say "self-adjoint", "Hermitian" or symmetric.

Theorem 15.4.2 (Normal ⇐⇒ diagonalizable with orthonormal basis)

Let V be a finite-dimensional complex inner product space. A linear map T : V → V is normal if and only if one can pick an orthonormal basis of eigenvectors.

Exercise 15.4.3. Show that if there exists such an orthonormal basis then T : V → V is normal, by writing T as a diagonal matrix in that basis.

Proof. This is long, and maybe should be omitted on a first reading. If T has an orthonormal basis of eigenvectors, this result is immediate. Now assume T is normal. We first prove T is diagonalizable; this is the hard part.

Claim 15.4.4. If T is normal, then ker T = ker T r = ker T † for r ≥ 1. (Here T r is T applied r times.)

Proof of Claim. Let S = T † • T , which is self-adjoint. We first note that S is Hermitian and ker S = ker T . To see it's Hermitian, note ⟨Sv, w⟩ = ⟨T v, T w⟩ = ⟨v, Sw⟩. Taking v = w also implies ker S ⊆ ker T (and hence equality since obviously ker T ⊆ ker S).

First, since we have S r (v), S r-2 (v) = S r-1 (v), S r-1 (v) , an induction shows that ker S = ker S r for r ≥ 1. Now, since T is normal, we have S r = (T † ) r • T r , and thus we have the inclusion ker T ⊆ ker T r ⊆ ker S r = ker S = ker T where the last equality follows from the first claim. Thus in fact ker T = ker T r . Finally, to show equality with ker

T † we ⟨T v, T v⟩ = v, T † T v = v, T T † v = T † v, T † v . ■

Now consider the given T , and any λ. In particular, for any eigenvalue λ of T , we find that ker(T -λid) = ker(T -λid) r . This implies that all the Jordan blocks of T have size 1; i.e. that T is in fact diagonalizable. Finally, we conclude that the eigenvectors of T and T † match, and the eigenvalues are complex conjugates. So, diagonalize T . We just need to show that if v and w are eigenvectors of T with distinct eigenvalues, then they are orthogonal. (We can use Gram-Schmidt on any eigenvalue that appears multiple times.) To do this, suppose T (v) = λv and T (w) = µw (thus T † (w) = µw). Then λ ⟨v, w⟩ = ⟨λv, w⟩ = ⟨T v, w⟩ = v, T † (w) = ⟨v, µw⟩ = µ ⟨v, w⟩ .

Since λ ̸ = µ, we conclude ⟨v, w⟩ = 0. This means that not only can we write A Hermitian matrix T is diagonalizable, and all its eigenvalues are real.

T =       λ 1 . . . . .

Proof. Obviously Hermitian =⇒ normal, so write it in the orthonormal basis of eigenvectors. To see that the eigenvalues are real, note that T = T † means λ i = λ i for every i. §15.5 A few harder problems to think about Problem 15A ⋆ (Double dual). Let V be a finite-dimensional vector space. Prove that

V → (V ∨ ) ∨ v → (ξ → ξ(v))

gives an isomorphism. (This is significant because the isomorphism is canonical, and in particular does not depend on the choice of basis. So this is more impressive.)

Problem 15B (Fundamental theorem of linear algebra). Let T : V → W be a map of finite-dimensional k-vector spaces. Prove that

dim im T = dim im T ∨ = dim V -dim ker T = dim W -dim ker T ∨ .

Problem 15C † (Row rank is column rank). A m × n matrix M of real numbers is given.

The column rank of M is the dimension of the span in R m of its n column vectors. The row rank of M is the dimension of the span in R n of its m row vectors. Prove that the row rank and column rank are equal.

Problem 15D (The complex conjugate spaces). Let V = (V, +, •) be a complex vector space. Define the complex conjugate vector space, denoted V = (V, +, * ) by changing just the multiplication:

c * v = c • v.

Show that for any sesquilinear form on V , if V is finite-dimensional, then

V → V ∨ v → ⟨-, v⟩

is an isomorphism of complex vector spaces.

Problem 15E (T † vs T ∨ ). Let V and W be real inner product spaces and let T : V → W be an inner product. Show that the following diagram commutes:

W V W ∨ V ∨ T † ∼ = ∼ = T ∨

Here the isomorphisms are v → ⟨-, v⟩. Thus, for real inner product spaces, T † is just T ∨ with the duals eliminated (by Theorem 15.2.2).

Problem 15F (Polynomial criteria for normality). Let V be a complex inner product space and let T : V → V be a linear map. Show that T is normal if and only if there is a polynomial

1 p ∈ C[t] such that T † = p(T ).

More on Groups

Part V: Contents 

17

Find all groups 207 17.1 Sylow theorems . 

18

The PID structure theorem 213 18.1 Finitely generated abelian groups . 

problems

Consider this problem from the 1996 AIME:

(AIME 1996) Two of the squares of a 7 × 7 checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible? What's happening here? Let X be the set of the 49 2 possible colorings of the board. What's the natural interpretation of "rotation"? Answer: the group Z/4Z = r | r 4 = 1 somehow "acts" on this set X by sending one state x ∈ X to another state r • x, which is just x rotated by 90 • . Intuitively we're just saying that two configurations are the same if they can be reached from one another by this "action".

We can make all of this precise using the idea of a group action. §16. 

• : G × X → X which lets a g ∈ G send an x ∈ X to g • x.

It satisfies the axioms (e) The group G can act on itself (i.e. X = G) by left multiplication: put g•g ′ := g⋆g ′ .

• (g 1 g 2 ) • x = g 1 • (g 2 • x) for any g 1 , g 2 ∈ G for all x ∈ X. • 1 G • x = x for any x ∈ X.

Exercise 16. 1.3 . Show that a group action can equivalently be described as a group homomorphism from G to S X , where S X is the symmetric group of permutations on X.

§16.2 Stabilizers and orbits

Prototypical example for this section: Again the AIME problem.

Given a group action G on X, we can define an equivalence relation ∼ on X as follows: x ∼ y if x = g • y for some g ∈ G. For example, in the AIME problem, ∼ means "one can be obtained from the other by a rotation".

Question 16.2.1. Why is this an equivalence relation?

In that case, the AIME problem wants the number of equivalence classes under ∼. So let's give these equivalence classes a name: orbits. We usually denote orbits by O.

As usual, orbits carve out X into equivalence classes.

X O 1 O 2 O 3

It turns out that a very closely related concept is:

Definition 16.2.2. The stabilizer of a point x ∈ X, denoted Stab G (x), is the set of g ∈ G which fix x; in other words Stab G (x) := {g ∈ G | g • x = x} .

Example 16.

2.3

Consider the AIME problem again, with X the possible set of states (again G = Z/4Z).

Let x be the configuration where two opposite corners are colored yellow. Evidently 1 G fixes x, but so does the 180 • rotation r 2 . But r and r 3 do not preserve x, so Stab G (x) = {1, r 2 } ∼ = Z/2Z. Once we realize the stabilizer is a group, this leads us to what I privately call the "fundamental theorem of how big an orbit is". In particular, the stabilizers of each x ∈ O have the same size.

Proof. The point is that every coset gS just specifies an element of O, namely g • x. The fact that S is a stabilizer implies that it is irrelevant which representative we pick. We know that G = Z/4Z acts on the set X of 49 2 possible coloring schemes. Now we can compute FixPt g explicitly for each g ∈ Z/4Z.

O ⊆ X S ⊆ G • • • • • g Since the |O| cosets partition G,

• If g = 1 G , then every coloring is fixed, for a count of 49 2 = 1176.

• If g = r 2 there are exactly 24 coloring schemes fixed by g: this occurs when the two squares are reflections across the center, which means they are preserved under a 180 • rotation.

• If g = r or g = r 3 , then there are no fixed coloring schemes.

As |G| = 4, the average is 1176 + 24 + 0 + 0 4 = 300. Let G = S 5 , and fix a π ∈ S 5 . Here's the question: is πσπ -1 related to σ? To illustrate this, I'll write out a completely random example of a permutation σ ∈ S 5 .

If σ = 1 → 3 2 → 1 3 → 5 4 → 2 5 → 4 then πσπ -1 = π(1) → π(3) π(2) → π(1) π(3) → π(5) π(4) → π(2) π(5) → π(4)

Thus our fixed π doesn't really change the structure of σ at all: it just "renames" each of the elements 1, 2, 3, 4, 5 to π(1), π(2), π(3), π(4), π(5). Problem 16F (Athemath Community-Building Event #1, Fall 2022). A group action • of a group G on set X is said to be • transitive if for every x 1 , x 2 ∈ X, there exists a g such that g • x 1 = x 2 ;

• faithful if the only element g ∈ G such that g • x = x for every x ∈ X is g = 1 G .

In other words, the only element which acts trivially on the entire set X is the identity element of G.

Does there exist a faithful transitive action of S 5 on a six-element set?

The following problem will hopefully never be proposed at the IMO.

Let n be a positive integer and let S = {1, . . . , n}. Find all functions f : S × S → S such that

(a) f (x, 1) = f (1, x) = x for all x ∈ S. (b) f (f (x, y), z) = f (x, f (y, z)) for all x, y, z ∈ S.

(c) For every x ∈ S there exists a y ∈ S such that f (x, y) = f (y, x) = 1.

Nonetheless, it's remarkable how much progress we've made on this "problem". In this chapter I'll try to talk about some things we have accomplished.

§17.1 Sylow theorems

Here we present the famous Sylow theorems, some of the most general results we have about finite groups. (c) Any two Sylow p-subgroups are conjugate subgroups (hence isomorphic).

Sylow's theorem is really huge for classifying groups; in particular, the conditions n p ≡ 1 (mod p) and n p | m can often pin down the value of n p to just a few values. Here are some results which follow from the Sylow theorems.

• A Sylow p-subgroup is normal if and only if n p = 1.

• Any group G of order pq, where p < q are primes, must have n q = 1, since n q ≡ 1 (mod q) yet n q | p. Thus G has a normal subgroup of order q.

• Since any abelian group has all subgroups normal, it follows that any abelian group has exactly one Sylow p-subgroup for every p dividing its order.

• If p ̸ = q, the intersection of a Sylow p-subgroup and a Sylow q-subgroup is just {1 G }. That's because the intersection of any two subgroups is also a subgroup, and Lagrange's theorem tells us that its order must divide both a power of p and a power of q; this can only happen if the subgroup is trivial.

Here's an example of another "practical" application.

Proposition 17. 1 

.2 (Triple product of primes)

If |G| = pqr is the product of distinct primes, then G must have a normal Sylow subgroup.

Proof. WLOG, assume p < q < r. Notice that n p ≡ 1 (mod p), n p |qr and cyclically, and assume for contradiction that n p , n q , n r > 1.

Since n r |pq, we have n r = pq since n r divides neither p nor q as n r ≥ 1 + r > p, q. Also, n p ≥ 1 + p and n q ≥ 1 + q. So we must have at least 1 + p Sylow p-subgroups, at least 1 + q Sylow q-subgroups, and at least pq Sylow r-subgroups.

But these groups are pretty exclusive. We claim that there are too many elements now. Indeed, if we count the non-identity elements contributed by these subgroups, we get n p (p -1) + n q (q -1) + n r (r -1) ≥ (1 + p)(p -1) + (1 + q)(q -1) + pq(r -1) > pqr which is more elements than G has! §17.

2 (Optional) Proving Sylow's theorem

The proof of Sylow's theorem is somewhat involved, and in fact many proofs exist. I'll present one below here. It makes extensive use of group actions, so I want to recall a few facts first. If G acts on X, then • The orbits of the action form a partition of X.

• if O is any orbit, then the orbit-stabilizer theorem says that

|O| = |G| / |Stab G (x)|

for any x ∈ O. 

H → gHg -1 = ghg -1 | h ∈ H .

If H and K are subgroups of G such that H = gKg -1 for some g ∈ G (in other words, they are in the same orbit under this action), then we say they are conjugate subgroups.

Because we somehow don't think of conjugate elements as "that different" (for example, in permutation groups), the following shouldn't be surprising: 

N G (H) := g ∈ G | gHg -1 = H .

In other words, it is the stabilizer of H under the conjugation action.

We are now ready to present the proof. • But the number of left cosets is m, which isn't divisible by p.

Hence some coset gP is a fixed point for every q, meaning qgP = gP for all q.

Equivalently, qg ∈ gP for all q ∈ Q, so Q ⊆ gP g -1 as desired. Now we can proceed with the proof. Let P act on S by conjugation. Then:

• Because P is a p-group, n p (mod p) is the number of fixed points of this action. Now we claim P is the only fixed point of this action.

• Let Q be any other fixed point, meaning xQx -1 = Q for any x ∈ P .

• Define the normalizer 

N G (Q) = g ∈ G | gQg -1 = Q . It

Z/12Z is 1 ⊴ Z/2Z ⊴ Z/4Z ⊴ Z/12Z.

Just like every integer breaks down as the product of primes, we can try to break every group down as a product of "basic" groups. Armed with our idea of quotient groups, the right notion is this. 

{1} ⊴ Z/2Z ⊴ Z/4Z ⊴ Z/12Z with factors Z/2Z, Z/2Z, Z/3Z {1} ⊴ Z/2Z ⊴ Z/6Z ⊴ Z/12Z with factors Z/2Z, Z/3Z, Z/2Z {1} ⊴ Z/3Z ⊴ Z/6Z ⊴ Z/12Z with factors Z/3Z, Z/2Z, Z/2Z.

These correspond to the factorization 12 = 2 2 • 3. This suggests that classifying all finite simple groups would be great progress, since every finite group is somehow a "product" of simple groups; the only issue is that there are multiple ways of building a group from constituents.

Amazingly, we actually have a full list of simple groups, but the list is really bizarre. Every finite simple group falls in one of the following categories:

• Z/pZ for p a prime,

• For n ≥ 5, the subgroup of S n consisting of "even" permutations.

• A simple group of Lie type (which I won't explain), and • Twenty-six "sporadic" groups which do not fit into any nice family.

The two largest of the sporadic groups have cute names. The baby monster group has order

2 41 • 3 13 • 5 6 • 7 2 • 11 • 13 • 17 • 19 • 23 • 31 • 47 ≈ 4 • 10 33

and the monster group (also "friendly giant") has order

2 46 • 3 20 • 5 9 • 7 6 • 11 2 • 13 3 • 17 • 19 • 23 • 29 • 31 • 41 • 47 • 59 • 71 ≈ 8 • 10 53 .

It contains twenty of the sporadic groups as subquotients (including itself), and these twenty groups are called the "happy family".

Math is weird. The main point of this chapter is to discuss a classification theorem for finitely generated abelian groups. This won't take long to do, and if you like, you can read just the first section and then move on.

However, since I'm here, I will go ahead and state the result as a special case of the much more general structure theorem. Its corollaries include • All finite-dimensional vector spaces are k ⊕n .

• The classification theorem for finitely generated abelian groups,

• The Jordan decomposition of a matrix from before,

• Another canonical form for a matrix: "Frobenius normal form". (c) Z ⊕2 is finitely generated (by two elements (1, 0) and (0, 1)).

(d) Z ⊕3 ⊕ Z/9Z ⊕ Z/2016Z is finitely generated by five elements.

(e) Z/3Z ⊕ Z/5Z is finitely generated by two elements. Let G be a finitely generated abelian group. Then there exists an integer r, prime powers q 1 , . . . , q m (not necessarily distinct) such that

G ∼ = Z ⊕r ⊕ Z/q 1 Z ⊕ Z/q 2 Z ⊕ • • • ⊕ Z/q m Z.

This decomposition is unique up to permutation of the Z/q i Z.

Definition 18.1.6. The rank of a finitely generated abelian group G is the integer r above. Now, we could prove this theorem, but it is more interesting to go for the gold and state and prove the entire structure theorem. §18.2 Some ring theory prerequisites Prototypical example for this section: R = Z.

Before I can state the main theorem, I need to define a few terms for UFD's, which behave much like Z:

Our intuition from the case R = Z basically carries over verbatim.

We don't even need to deal with prime ideals and can factor elements instead.

Definition 18.2.1. If R is a UFD, then p ∈ R is a prime element if (p)

is a prime ideal and p ̸ = 0. For UFD's this is equivalent to: if p = xy then either x or y is a unit.

So for example in Z the set of prime elements is {±2, ±3, ±5, . . . }. Now, since R is a UFD, every element r factors into a product of prime elements

r = up e 1

1 p e 2 2 . . . p em m Definition 18.2.2. We say r divides s if s = r ′ r for some r ′ ∈ R. This is written r | s.

Example 18.2.3 (Divisibility in Z)

The number 0 is divisible by every element of Z. All other divisibility as expected.

Question 18.2.4. Show that r | s if and only if the exponent of each prime in r is less than or equal to the corresponding exponent in s.

Now, the case of interest is the even stronger case when R is a PID: Proposition 18.2.5 (PID's are Noetherian UFD's)

If R is a PID, then it is Noetherian and also a UFD.

Proof. The fact that R is Noetherian is obvious. For R to be a UFD we essentially repeat the proof for Z, using the fact that (a, b) is principal in order to extract gcd(a, b).

In this case, we have a Chinese remainder theorem for elements.

Theorem 18.2.6 (Chinese remainder theorem for rings)

Let m and n be relatively prime elements, meaning (m)

+ (n) = (1). Then R/(mn) ∼ = R/(m) × R/(n).

Here the ring product is as defined in Example 4. 3.8 .

Proof. This is the same as the proof of the usual Chinese remainder theorem. First, since (m, n) = (1) we have am + bn = 1 for some a and b. Then we have a map

R/(m) × R/(n) → R/(mn) by (r, s) → r • bn + s • am.

One can check that this map is well-defined and an isomorphism of rings. (Diligent readers invited to do so.)

Finally, we need to introduce the concept of a Noetherian R-module.

Definition 18.2.7. An R-module M is Noetherian if it satisfies one of the two equivalent conditions:

• Its submodules obey the ascending chain condition: there is no infinite sequence of modules

M 1 ⊊ M 2 ⊊ . . . . • All submodules of M (including M itself) are finitely generated.

This generalizes the notion of a Noetherian ring: a Noetherian ring R is one for which R is Noetherian as an R-module. Let R be a PID and let M be any finitely generated R-module. Then

M ∼ = m i=1 R/(s i ) for some s i (possibly zero) satisfying s 1 | s 2 | • • • | s m .

Corollary 18.3.2 (Structure theorem, primary form)

Let R be a PID and let M be any finitely generated R-module. Then

M ∼ = R ⊕r ⊕ R/(q 1 ) ⊕ R/(q 2 ) ⊕ • • • ⊕ R/(q m )

where q i = p e i i for some prime element p i and integer e i ≥ 1.

Proof of corollary.

Factor each s i into prime factors (since R is a UFD), then use the Chinese remainder theorem.

Remark 18.3.3 -In both theorems the decomposition is unique up to permutations of the summands; good to know, but I won't prove this.

§18.4 Reduction to maps of free R-modules

Definition 18.4.1. A free R-module is a module of the form R ⊕n (or more generally, I R for some indexing set I, just to allow an infinite basis).

The proof of the structure theorem proceeds in two main steps. First, we reduce the problem to a linear algebra problem involving free R-modules R ⊕d . Once that's done, we just have to play with matrices; this is done in the next section.

Suppose M is finitely generated by d elements. Then there is a surjective map of R-modules R ⊕d ↠ M whose image on the basis of R ⊕d are the generators of M . Let K denote the kernel.

We claim that K is finitely generated as well. To this end we prove that Lemma 18.4.2 (Direct sum of Noetherian modules is Noetherian)

Let M and N be two Noetherian R-modules. Then the direct sum M ⊕ N is also a Noetherian R-module.

Proof. It suffices to show that if L ⊆ M ⊕ N , then L is finitely generated. One guess is that L = P ⊕ Q, where P and Q are the projections of L onto M and N . Unfortunately this is false (take

M = N = Z and L = {(n, n) | n ∈ Z}

) so we will have to be more careful.

Consider the submodules

A = {x ∈ M | (x, 0) ∈ L} ⊆ M B = {y ∈ N | ∃x ∈ M : (x, y) ∈ L} ⊆ N.

(Note the asymmetry for A and B: the proof doesn't work otherwise.) Then A is finitely generated by a 1 , . Hence K ⊆ R ⊕d is finitely generated as claimed. So we can find another surjective map

R ⊕f ↠ K. Consequently, we have a composition K R ⊕f R ⊕d M ⊃ T

Observe that M is the cokernel of the linear map T , i.e. we have that

M ∼ = R ⊕d / im(T ).

So it suffices to understand the map T well.

§18.5 Smith normal form

The idea is now that we have reduced our problem to studying linear maps T : R ⊕m → R ⊕n , which can be thought of as a generic matrix Of course, as you might expect it ought to be possible to change the given basis of T such that T has a nicer matrix form. We already saw this in Jordan form, where we had a map T : V → V and changed the basis so that T was "almost diagonal". This time, we have two sets of bases we can change, so we would hope to get a diagonal basis, or even better.

T =   

Before proceeding let's think about how we might edit the matrix: what operations are permitted? Here are some examples:

• Swapping rows and columns, which just corresponds to re-ordering the basis.

• Adding a multiple of a column to another column. For example, if we add 3 times the first column to the second column, this is equivalent to replacing the basis (e 1 , e 2 , e 3 , . . . , e m ) → (e 1 , e 2 + 3e 1 , e 3 , . . . , e m ).

• Adding a multiple of a row to another row. One can see that adding 3 times the first row to the second row is equivalent to replacing the basis

(f 1 , f 2 , f 3 , . . . , f n ) → (f 1 -3f 2 , f 2 , f 3 , . . . , f n ).

More generally,

If A is an invertible n × n matrix we can replace T with AT .

This corresponds to replacing

(f 1 , . . . , f n ) → (A -1 (f 1 ), . . . , A -1 (f n ))

(the "invertible" condition just guarantees the latter is a basis). Of course similarly we can replace X with XB where B is an invertible m × m matrix; this corresponds to

(e 1 , . . . , e m ) → (B -1 (e 1 ), . . . , B -1 (e m ))

Armed with this knowledge, we can now approach:

Theorem 18.5.1 (Smith normal form)

Let R be a PID. Let M = R ⊕m and N = R ⊕n be free R-modules and let T : M → N be a linear map. Set k = min{m, n}.

Then we can select a pair of new bases for M and N such that T has only diagonal entries s 1 , s 2 , . . . , s k and

s 1 | s 2 | • • • | s k .

So if m > n, the matrix should take the form

      s 1 0 0 0 . . . 0 0 s 2 0 0 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 0 s n . . . 0      

.

and similarly when m ≤ n.

Question 18.5.2. Show that Smith normal form implies the structure theorem.

Remark 18.5.3 -Note that this is not a generalization of Jordan form.

• In Jordan form we consider maps T : V → V ; note that the source and target space are the same, and we are considering one basis for the space V .

• In Smith form the maps T : M → N are between different modules, and we pick two sets of bases (one for M and one for N ).

Example 18.5.4 (Example of Smith normal form)

To give a flavor of the idea of the proof, let's work through a concrete example with the Z-matrix 18 38 48 14 30 32 .

The GCD of all the entries is 2, and so motivated by this, we perform the Euclidean algorithm on the left column: subtract the second row from the first row, then three times the first row from the second: Now that the GCD of 2 is present, we move it to the upper-left by switching the two rows, and then kill off all the entries in the same row/column; since 2 was the GCD all along, we isolate 2 completely:

18

4 8 16 2 6 -16 → 2 6 -16 4 8 16 → 2 6 -16 0 -4 48 → 2 0 0 0 -4 48 .

This reduces the problem to a 1 × 2 matrix. So we just apply the Euclidean algorithm again there: We want to show, by a series of operations (gradually changing the given basis) that we can rearrange the matrix into Smith normal form. Define gcd(x, y) to be any generator of the principal ideal (x, y).

2 0 0 0 -4 0 → 2 0 0 0 4 0 .

Claim 18.5.5 ("Euclidean algorithm"). If a and b are entries in the same row or column, we can change bases to replace a with gcd(a, b) and b with something else.

Proof. We do just the case of columns. By hypothesis, gcd(a, b) = xa + yb for some x, y ∈ R. We must have (x, y) = ( 1) now (we're in a UFD). So there are u and v such that

xu + yv = 1. Then x y -v u a b = gcd(a, b) something

and the first matrix is invertible (check this!), as desired. ■

Let s 1 = (a ij ) i,j be the GCD of all entries. Now by repeatedly applying this algorithm, we can cause s to appear in the upper left hand corner. Then, we use it to kill off all the entries in the first row and the first column, thus arriving at a matrix

        s 1 0 0 . . . 0 0 a ′ 22 a ′ 23 . . . a ′ 2n 0 a ′ 32 a ′ 33 . . . a ′ 3n . . . . . . . . . . . . . . . 0 a ′ m2 a ′ m3 . . . a ′ mn        

. Now we repeat the same procedure with this lower-right (m -1) × (n -1) matrix, and so on. This gives the Smith normal form.

With the Smith normal form, we have in the original situation that

M ∼ = R ⊕d / im T

and applying the theorem to T completes the proof of the structure theorem. §18.6 A few harder problems to think about Now, we can apply our structure theorem! Problem 18A † (Finite-dimensional vector spaces are all isomorphic). A vector space V over a field k has a finite spanning set of vectors. Show that V ∼ = k ⊕n for some n.

Problem 18B † (Frobenius normal form). Let T : V → V where V is a finite-dimensional vector space over an arbitrary field k (not necessarily algebraically closed). Show that one can write T as a block-diagonal matrix whose blocks are all of the form

        0 0 0 . . . 0 * 1 0 0 . . . 0 * 0 1 0 . . . 0 * . . . . . . . . . . . . . . . . . . 0 0 0 . . . 1 *         . (View V as a k[x]-module with action x • v = T (v).)

Problem 18C † (Jordan normal form). Let T : V → V where V is a finite-dimensional vector space over an arbitrary field k which is algebraically closed. Prove that T can be written in Jordan form. 

20

Semisimple algebras 20.1 Schur's lemma continued . 

21.2

The dual space modulo the commutator . 

22

Some applications 22.1 Frobenius divisibility .

In the 19th century, the word "group" hadn't been invented yet; all work was done with subsets of GL(n) or S n . Only much later was the abstract definition of a group was given, an abstract set G which was an object in its own right.

While this abstraction is good for some reasons, it is often also useful to work with concrete representations. This is the subject of representation theory. Linear algebra is easier than abstract algebra, so if we can take a group G and represent it concretely as a set of matrices in GL(n), this makes them easier to study. This is the representation theory of groups: how can we take a group and represent its elements as matrices? §19.1 Algebras Prototypical example for this section: k[x 1 , . . 

. , x n ] and k[G].

Rather than working directly with groups from the beginning, it will be more convenient to deal with so-called k-algebras. This setting is more natural and general than that of groups, so once we develop the theory of algebras well enough, it will be fairly painless to specialize to the case of groups.

Colloquially, (c) The set of n × n matrices with entries in k, which we denote by Mat n (k). Note the multiplication here is not commutative.

An

(d) The set Mat(V ) of linear operators T : V → V , with multiplication given by the composition of operators. (Here V is some vector space over k.) This is really the same as the previous example.

Definition 19.1.2. Let k be a field. A k-algebra A is a possibly noncommutative ring, equipped with a ring homomorphism k → A, whose image is the "copy of k". (In particular,

1 k → 1 A .)

Thus we can consider k as a subset of A, and we then additionally require λ • a = a • λ for each λ ∈ k and a ∈ A.

If the multiplication operation is also commutative, then we say A is a commutative algebra.

Definition 19. 1.3 . Equivalently, a k-algebra A is a k-vector space which also has an associative, bilinear multiplication operation (with an identity 1 A ). The "copy of k" is obtained by considering elements λ1 A for each λ ∈ k (i.e. scaling the identity by the elements of k, taking advantage of the vector space structure).

Abuse of Notation 19. 1.4 . Some other authors don't require A to be associative or to have an identity, so to them what we have just defined is an "associative algebra with 1". However, this is needlessly wordy for our purposes.

Example 19. 1 

.5 (Group algebra)

The group algebra k[G] is the k-vector space whose basis elements are the elements of a group G, and where the product of two basis elements is the group multiplication. For example, suppose

G = Z/2Z = {1 G , x}. Then k[G] = {a1 G + bx | a, b ∈ k} with multiplication given by (a1 G + bx)(c1 G + dx) = (ac + bd)1 G + (bc + ad)x. Question 19.1.6. When is k[G] commutative?

The example k[G] is very important, because (as we will soon see) a representation of the algebra k[G] amounts to a representation of the group G itself.

It is worth mentioning at this point that: (i) A k-vector space V , and (ii) An action • of A on V : thus, for every a ∈ A we can take v ∈ V and act on it to get a • v. This satisfies the usual axioms:

• (a + b) • v = a • v + b • v, a • (v + w) = a • v + a • w, and (ab) • v = a • (b • v). • λ • v = λv for λ ∈ k. In particular, 1 A • v = v.

Definition 19.2.2. The action of A can be more succinctly described as saying that there is a k-algebra homomorphism ρ :

A → Mat(V ). (So a • v = ρ(a)(v).

) Thus we can also define a representation of A as a pair

(V, ρ : A → Mat(V )) .

This is completely analogous to how a group action G on a set X with n elements just amounts to a group homomorphism G → S n . From this perspective, what we are really trying to do is:

If A is an algebra, we are trying to represent the elements of A as matrices.

Abuse of Notation 19.2.3. While a representation is a pair (V, ρ) of both the vector space V and the action ρ, we frequently will just abbreviate it to "V ". This is probably one of the worst abuses I will commit, but everyone else does it and I fear the mob.

Abuse of Notation 19.2.4. Rather than ρ(a)(v) we will just write ρ(a)v.

Example 19.2.5 (Representations of Mat

(V )) (a) Let A = Mat 2 (R). Then there is a representation (R ⊕2 , ρ) where a matrix a ∈ A just acts by a • v = ρ(a)(v) = a(v).

(b) More generally, given a vector space V over any field k, there is an obvious

representation of A = Mat(V ) by a • v = ρ(a)(v) = a(v) (since a ∈ Mat(V )).

From the matrix perspective: if A = Mat(V ), then we can just represent A as matrices over V .

(c) There are other representations of A = Mat 2 (R). A silly example is the representation (R ⊕4 , ρ) given by

ρ : a b c d →      a b 0 0 c d 0 0 0 0 a b 0 0 c d      . More abstractly, viewing R ⊕4 as (R ⊕2 ) ⊕ (R ⊕2 ), this is a • (v 1 , v 2 ) = (a • v 1 , a • v 2 ). Example 19.2.6 (Representations of polynomial algebras) (a) Let A = k. Then a representation of k is just any k-vector space V . (b) If A = k[x]

, then a representation (V, ρ) of A amounts to a vector space V plus the choice of a linear operator

T ∈ Mat(V ) (by T = ρ(x)). (c) If A = k[x]/(x 2

) then a representation (V, ρ) of A amounts to a vector space V plus the choice of a linear operator T ∈ Mat(V ) satisfying T 2 = 0.

(d) We can create arbitrary "functional equations" with this pattern. For example, if A = k[x, y]/(x 2 -x + y, y 4 ) then representing A by V amounts to finding commuting operators S, T ∈ Mat(V ) satisfying S 2 = S -T and T 4 = 0.

Example 19.2.7 (Representations of groups)

(a) Let A = R[S 3 ]. Then let V = R ⊕3 = {(x, y, z) | x, y, z ∈ R}.

We can let A act on V as follows: given a permutation π ∈ S 3 , we permute the corresponding coordinates in V . So for example, if

If π = (1 2) then π • (x, y, z) = (y, x, z).

This extends linearly to let A act on V , by permuting the coordinates.

From the matrix perspective, what we are doing is representing the permutations in S 3 as permutation matrices on k ⊕3 , like 

(1 2) →    0 1 0 1 0 0 0 0 1    . (b) More generally, let A = k[G]. Then a representation (V, ρ) of A amounts to a group homomorphism ψ : G → GL(V ). (In particular, ρ(1 G ) = id V .)

V = (V, ρ V ) and W = (W, ρ W ) be two representations of A. Then V ⊕ W is a representation, with action ρ given by a • (v, w) = (a • v, a • w).

This representation is called the direct sum of V and W .

Example 19.3.2

Earlier we let Mat 2 (R) act on R ⊕4 by

ρ : a b c d →      a b 0 0 c d 0 0 0 0 a b 0 0 c d      .

So this is just a direct sum of two two-dimensional representations.

You can also view the vectors of R ⊕4 as two vectors in R ⊕2 "stacked horizontally" as e f g h , so the action would be given by

a b c d • e f g h = ae + bg af + bh ce + dg cf + dh .

Remark 19.3.3 -Perhaps this is the reason why people tend to write V as the representation without the accompanied ρ V , as long as it's possible to embed the k-algebra A into a subalgebra of Mat d (k), then V can be isomorphically embedded as a subrepresentation of (k ⊕d ) ⊕m , being m copied of the obvious k ⊕d representation stacked horizontally.

More generally, given representations (V, ρ V ) and (W,

ρ W ) the representation ρ of V ⊕ W looks like ρ(a) = ρ V (a) 0 0 ρ W (a) .

Example 19.3.4 (Representation of S n decomposes)

Let A = R[S 3 ] again, acting via permutation of coordinates on

V = R ⊕3 = {(x, y, z) | x, y, z ∈ R}.

Consider the two subspaces

W 1 = {(t, t, t) | t ∈ R} W 2 = {(x, y, z) | x + y + z = 0} . Note V = W 1 ⊕ W 2 as vector spaces. But each of W 1 and W 2 is a subrepresentation (since the action of A keeps each W i in place), so V = W 1 ⊕ W 2 as representations too.

Direct sums also come up when we play with algebras.

Proposition 19.3.5 (Representations of A ⊕ B are V A ⊕ V B )

Let A and B be k-algebras. Then every representation of A ⊕ B is of the form

V A ⊕ V B

where V A and V B are representations of A and B, respectively.

Example 19.3.6

Take A = B = Mat 2 (R). There are two obvious representations of the k-algebra A ⊕ B, V A and V B , corresponds to the action of A and B respectively. Each of V A and V B are isomorphic to R ⊕2 as R-vector spaces.

What this proposition says is that, you cannot "mix" the action of A and B in order to get some representation

V ∼ = R 2 of A ⊕ B, such as by (a + b) • v = a • v + 2b • v for a ∈ A and b ∈ B. Sketch of Proof. Let (V, ρ) be a representation of A ⊕ B. For any v ∈ V , ρ(1 A + 1 B )v = ρ(1 A )v + ρ(1 B )v. One can then set V A = {ρ(1 A )v | v ∈ V } and V B = {ρ(1 B )v | v ∈ V }. These are disjoint, since if ρ(1 A )v = ρ(1 B )v ′ , we have ρ(1 A )v = ρ(1 A 1 A )v = ρ(1 A 1 B )v ′ = 0 V ,

and similarly for the other side.

In the example above, if you see the representation V

A ⊕ V B as R 4 , then any element in A acting on an element in V A ⊕ V B would zero out the V B -component of the vector.

So, the key idea of the proof is:

The A and B component of A ⊕ B is used to act on V , in order to project the vector space V into the components V A and V B to separate out the subrepresentations.

§19.4 Irreducible and indecomposable representations

Prototypical example for this section: k[S 3 ] decomposes as the sum of two spaces.

One of the goals of representation theory will be to classify all possible representations of an algebra A. If we want to have a hope of doing this, then we want to discard "silly" representations such as

ρ : a b c d →      a b 0 0 c d 0 0 0 0 a b 0 0 c d     

and focus our attention instead on "irreducible" representations. This motivates:

Definition 19.4.1. Let V be a representation of A. A subrepresentation W ⊆ V is a subspace W with the property that for any a ∈ A and w ∈ W , a • w ∈ W . In other words, this subspace is invariant under actions by A. 

Thus for example if

V = W 1 ⊕ W 2 for representations W 1 , W 2 then W 1 and W 2 are subrepresentations of V . Definition 19.4.2. If V has no proper nonzero subrepresentations then it is irreducible. If there is no pair of proper subrepresentations W 1 , W 2 such that V = W 1 ⊕ W 2 ,

V = R ⊕3 = {(x, y, z) | x, y, z ∈ R}.

Consider again the two subspaces

W 1 = {(t, t, t) | t ∈ R} W 2 = {(x, y, z) | x + y + z = 0} .

As we've seen, V = W 1 ⊕ W 2 , and thus V is not irreducible. But one can show that W 1 and W 2 are irreducible (and hence indecomposable) as follows.

• For W 1 it's obvious, since W 1 is one-dimensional.

• For W 2 , consider any vector w = (a, b, c) with a + b + c = 0 and not all zero.

Then WLOG we can assume a ̸ = b (since not all three coordinates are equal).

In that case, (1 2) sends w to w ′ = (b, a, c). Then w and w ′ span W 2 .

Thus V breaks down completely into irreps.

Unfortunately, if W is a subrepresentation of V , then it is not necessarily the case that we can find a supplementary vector space W ′ such that V = W ⊕ W ′ . Put another way, if V is reducible, we know that it has a subrepresentation, but a decomposition requires two subrepresentations. Here is a standard counterexample: 

Show that the only subrepresentation is

W = {(t, 0) | t ∈ R}. So V is not irreducible, but it is indecomposable.

Here is a slightly more optimistic example, and the "prototypical example" that you should keep in mind. 

: V → W such that T (a • v) = a • T (v)

for any a ∈ A, v ∈ V . (Note that the first • is the action of ρ V and the second • is the action of ρ W .) This is exactly what you expect if you think that V and W are "left A-modules". If T is invertible, then it is an isomorphism of representations and we say

V ∼ = W . Remark 19.5.2 (For commutative diagram lovers) -The condition T (a • v) = a • T (v) can be read as saying that V V W W ρ 1 (a) T T ρ 2 (a)

commutes for any a ∈ A.

Remark 19.5.3 (For category lovers) -A representation is just a "bilinear" functor from an abelian one-object category { * } (so Hom( * , * ) ∼ = A) to the abelian category Vect k . Then an intertwining operator is just a natural transformation.

Here are some examples of intertwining operators.

Example 19.5.4 (Intertwining operators) (a) For any λ ∈ k, the scalar map

T (v) = λv is intertwining. (b) If W ⊆ V is a subrepresentation, then the inclusion W → V is an intertwining operator. (c) The projection map V 1 ⊕ V 2 ↠ V 1 is an intertwining operator. (d) Let V = R ⊕2 and represent A = k[x] by (V, ρ) where ρ(x) = 0 1 -1 0 .

Thus ρ(x) is rotation by 90 • around the origin. Let T be rotation by 30 • . Then T : V → V is intertwining (the rotations commute).

Example 19.5.5 (A non-example: Representation of Mat(V ))

Let A = Mat 2 (R) ⊕ Mat 2 (R).

Then A can be viewed as a subset of Mat 4 (R) of the matrices of the form

     a b 0 0 c d 0 0 0 0 e f 0 0 g h     

.

There are two obvious irreps of A, given by V 1 consisting of the vectors in R 4 of the form (m, n, 0, 0), and V 2 consisting of the vectors in R 4 of the form (0, 0, p, q). In this case, even though V 1 and V 2 are isomorphic as R-vector spaces, they're not isomorphic as representations of A -so any intertwining operator from V 1 to V 2 must be identically zero.

Exercise 19.5.6 (Kernel and image are subrepresentations). Let T : V → W be an intertwining operator.

(a) Show that ker

T ⊆ V is a subrepresentation of V . (b) Show that im T ⊆ W is a subrepresentation of W .

The previous exercise gives us the famous Schur's lemma.

Theorem 19.5.7 (Schur's lemma)

Let V and W be representations of a k-algebra A. Let T : V → W be a nonzero intertwining operator. Then (a) If V is irreducible, then T is injective. (b) If W is irreducible, then T is surjective.

In particular if both V and W are irreducible then T is an isomorphism.

An important special case is if k is algebraically closed: then the only intertwining operators T : V → V are multiplication by a constant.

Theorem 19.5.8 (Schur's lemma for algebraically closed fields)

Let k be an algebraically closed field. Let V be an irrep of a k-algebra A. Then any intertwining operator T : V → V is multiplication by a scalar. We have already seen the counterexample of rotation by 90 • for k = R; this was the same counterexample we gave to the assertion that all linear maps have eigenvalues.

§19.6 The representations of Mat d (k)

To give an example of the kind of progress already possible, we prove:

Theorem 19.6.1 (Representations of Mat d (k))

Let k be any field, d be a positive integer and let W = k ⊕d be the obvious representation of A = Mat d (k). Then the only finite-dimensional representations of Mat d (k) are W ⊕n for some positive integer n (up to isomorphism). In particular, it is irreducible if and only if n = 1.

For concreteness, I'll just sketch the case d = 2, since the same proof applies verbatim to other situations. This shows that the examples of representations of Mat 2 (R) we gave earlier are the only ones.

As we've said this is essentially a functional equation. The algebra A = Mat 2 (k) has basis given by four matrices

E 1 = 1 0 0 0 , E 2 = 0 0 0 1 , E 3 = 0 1 0 0 , E 4 = 0 0 1 0 satisfying relations like E 1 + E 2 = id A , E 2 i = E i , E 1 E 2 = 0, etc.

So let V be a representation of A, and let M i = ρ(E i ) for each i; we want to classify the possible matrices M i on V satisfying the same functional equations. This is because, for example,

id V = ρ(id A ) = ρ(E 1 + E 2 ) = M 1 + M 2 .

By the same token M 1 M 3 = M 3 . Proceeding in a similar way, we can obtain the following multiplication table:

× M 1 M 2 M 3 M 4 M 1 M 1 0 M 3 0 M 2 0 M 2 0 M 4 M 3 0 M 3 0 M 1 M 4 M 4 0 M 2 0 and M 1 + M 2 = id V

Note that each M i is a linear operator V → V ; for all we know, it could have hundreds of entries. Nonetheless, given the multiplication table of the basis E i we get the corresponding table for the M i . So, in short, the problem is as follows:

Find all vector spaces V and quadruples of matrices M i satisfying the multiplication table above.

Let

W 1 = M img 1 (V ) and W 2 = M img 2 (V ) be the images of M 1 and M 2 . Claim 19.6.2. V = W 1 ⊕ W 2 .

Proof. First, note that for any v ∈ V we have

v = ρ(id)(v) = (M 1 + M 2 )v = M 1 v + M 2 v.

Moreover, we have that

W 1 ∩ W 2 = {0}, because if M 1 v 1 = M 2 v 2 then M 1 v 1 = M 1 (M 1 v 1 ) = M 1 (M 2 v 2 ) = 0. Claim 19.6.3. W 1 ∼ = W 2 .

Proof. Check that the maps

W 1 ×M 4 ---→ W 2 and W 2 ×M 3 ---→ W 1

are well-defined and mutually inverse. Now, let e 1 , . . . , e n be basis elements of W 1 ; thus M 4 e 1 , . . . , M 4 e n are basis elements of W 2 . However, each {e j , M 4 e j } forms a basis of a subrepresentation isomorphic to

W = k ⊕2 (what's the isomorphism?).

This finally implies that all representations of A are of the form W ⊕n . In particular, W is irreducible because there are no representations of smaller dimension at all! §19.7 A few harder problems to think about Problem 19E. Let (V, ρ) be an indecomposable (not irreducible) representation of an algebra A. Prove that any intertwining operator T : V → V is either nilpotent or an isomorphism.

Problem 19A † . Suppose we have one-dimensional representations V 1 = (V 1 , ρ 1 ) and V 2 = (V 2 , ρ 2 ) of A. Show that V 1 ∼ = V 2 if and only if ρ 1 (a)

(Note that Theorem 19.5.8 doesn't apply, since the field k may not be algebraically closed.)

20 Semisimple algebras

In what follows, assume the field k is algebraically closed. Fix an algebra A and suppose you want to study its representations. We have a "direct sum" operation already. So, much like we pay special attention to prime numbers, we're motivated to study irreducible representations and then build all the representations of A from there.

Unfortunately, we have seen (Exercise 19. 4.5 ) that there exists a representation which is not irreducible, and yet cannot be broken down as a direct sum (indecomposable). This is weird and bad, so we want to give a name to representations which are more well-behaved. We say that a representation is completely reducible if it doesn't exhibit this bad behavior.

Even better, we say a finite-dimensional algebra A is semisimple if all its finitedimensional representations are completely reducible. So when we study finite-dimensional representations of semisimple algebras A, we just have to figure out what the irreps are, and then piecing them together will give all the representations of A.

In fact, semisimple algebras A have even nicer properties. The culminating point of the chapter is when we prove that A is semisimple if and only if A ∼ = i Mat(V i ), where the V i are the irreps of A (yes, there are only finitely many!).

In the end, we will see that the group algebras k[G] of a finite group G are all semisimple (at least when k has characteristic 0), thus we're justified in focusing on studying the semisimple algebras.

Remark 20.0.1 (Digression) -The converse does not hold, however -if k has characteristic 0, not every finite-dimensional semisimple k-algebra is isomorphic to some group algebra. Classifying exactly when a k-algebra is isomorphic to a group algebra turns out to be a hard question, see https://mathoverflow.net/q/314502. §20.1 Schur's lemma continued Prototypical example for this section:

For V irreducible, Hom rep (V ⊕2 , V ⊕2 ) ∼ = k ⊕4 .

Definition 20.1.1. For an algebra A and representations V and W , we let Hom rep (V, W ) be the set of intertwining operators between them. (It is also a k-algebra.) By Schur's lemma (since k is algebraically closed, which again, we are taking as a standing assumption), we already know that if V and W are irreps, then

Hom rep (V, W ) ∼ = k if V ∼ = W 0 if V ̸ ∼ = W .

Can we say anything more? For example, it also tells us that

Hom rep (V, V ⊕2 ) = k ⊕2 . The possible maps are v → (c 1 v 1 , c 2 v 2 ) for some choice of c 1 , c 2 ∈ k.

More generally, suppose V is an irrep and consider Hom rep (V ⊕m , V ⊕n ). Intertwining operators T : V ⊕m → V ⊕n are determined completely by the mn choices of compositions

V V ⊕m V ⊕n V ⊃ T

where the first arrow is inclusion to the ith component of V ⊕m (for 1 ≤ i ≤ m) and the second arrow is inclusion to the jth component of V ⊕n (for 1 ≤ j ≤ n). However, by Schur's lemma on each of these compositions, we know they must be constant.

Thus, Hom rep (V ⊕n , V ⊕m ) consist of n × m "matrices" of constants, and the map is provided by

      c 11 c 12 . . . c 1(n-1) c 1n c 21 c 22 . . . c 2(n-1) c 2n . . . . . . . . . . . . . . . c m1 c m2 . . . c m(n-1) c mn             v 1 v 2 . . . v n       ∈ V ⊕m

where the c ij ∈ k but v i ∈ V ; note the type mismatch! This is not just a k-linear map V ⊕n → V ⊕m ; rather, the outputs are m linear combinations of the inputs.

More generally, we have:

Theorem 20.1.2 (Schur's lemma for completely reducible representations)

Let V and W be completely reducible representations, and set

V = V ⊕n i i , W = V ⊕m i i for integers n i , m i ≥ 0, where each V i is an irrep. Then Hom rep (V, W ) ∼ = i Mat m i ×n i (k)

meaning that an intertwining operator T : V → W amounts to, for each i, an m i × n i matrix of constants which gives a map

V ⊕n i i → V ⊕m i i .

Corollary 20. 1 

.3 (Subrepresentations of completely reducible representations)

Let V = V ⊕n i i be completely reducible. Then any subrepresentation W of V is isomorphic to V ⊕m i i

where m i ≤ n i for each i, and the inclusion W → V is given by the direct sum of inclusion

V ⊕m i i → V ⊕n i i , which are n i × m i matrices.

Proof. Apply Schur's lemma to the inclusion W → V .

Recall from Section 9.5 that a linear maps from a n-dimensional vector space to a mdimensional vector space can be written as a n × m matrix. Here the situation is similar, however the matrices are made for each irrep independently, and the non-isomorphic irreps, in some sense, "doesn't talk to each other".

Remark 20.1.4 -The representation V ⊕n can also be viewed as n vectors of V "stacked horizontally", as we did in Example 19.3.2:

    . . . . . . . . . v 1 v 2 • • • v n . . . . . . . . .     ∈ V ⊕n .

That way, the action is given by

    . . . . . . . . . v 1 v 2 • • • v n . . . . . . . . .           c 11 c 21 • • • c (m-1)1 c m1 c 12 c 22 • • • c (m-1)1 c m2 . . . . . . . . . . . . . . . c 1n c 2n • • • c (m-1)n c mn       ∈ V ⊕m .

It may be clearer this way to see the type mismatch happening. And this also gives a natural explanation why the intertwining operators in Problem 19D ⋆ corresponds to right matrix multiplication. §20.

2 Density theorem

We are going to take advantage of the previous result to prove that finite-dimensional algebras have finitely many irreps.

Theorem 20.2.1 (Jacobson density theorem)

Let (V 1 , ρ 1 ), . . . , (V r , ρ r ) be pairwise nonisomorphic irreps of A.

Then there is a surjective map of vector spaces

r i=1 ρ i : A ↠ r i=1 Mat(V i ).

The right way to think about this theorem is that Density is the "Chinese remainder theorem" for irreps of A.

Recall that in number theory, the Chinese remainder theorem tells us that given lots of "unrelated" congruences, we can find a single N which simultaneously satisfies them all. Similarly, given lots of different nonisomorphic irreps of A, this means that we can select a single a ∈ A which induces any tuple (ρ 1 (a), . . . , ρ r (a)) of actions we want -a surprising result, since even the r = 1 case is not obvious at all!

ρ 1 (a) = M 1 ∈ Mat(V 1 ) ρ 2 (a) = M 2 ∈ Mat(V 2 ) a ∈ A . . . ρ r (a) = M r ∈ Mat(V r )

This also gives us the non-obvious corollary:

Corollary 20.

2.2 (Finiteness of number of representations)

Any finite-dimensional algebra A has at most dim A irreps.

Proof.

If V i are such irreps then A ↠ i V ⊕ dim V i i , hence we have the inequality (dim V i ) 2 ≤ dim A. Proof of density theorem. Let V = V 1 ⊕ • • • ⊕ V r , so A acts on V = (V, ρ) by ρ = i ρ i .

Thus by Problem 19C ⋆ , we can instead consider ρ as an intertwining operator

ρ : Reg(A) → r i=1 Mat(V i ) ∼ = r i=1 V ⊕d i i .

We will use this instead as it will be easier to work with.

First, we handle the case r = 1. Fix a basis e 1 , . . . , e n of V = V 1 . Assume for contradiction that the map is not surjective. Then there is a map of representations (by ρ and the isomorphism) Reg(A) → V ⊕n given by a → (a • e 1 , . . . , a • e n ). By hypothesis, it is not surjective: its image is a proper subrepresentation of V ⊕n . Assume its image is isomorphic to V ⊕m for m < n, so by Theorem 20. 1.2 there is a matrix of constants X with

Reg(A) V ⊕n V ⊕m a (a • e 1 , . . . , a • e n ) 1 A (e 1 , . . . , e n ) (v 1 , . . . , v m ) ⊃ X•-

where the two arrows in the top row have the same image; hence the pre-image (v 1 , . . . , v m ) of (e 1 , . . . , e n ) can be found. But since m < n we can find constants c 1 , . . . , c n not all zero such that X applied to the column vector (c 1 , . . . , c n ) is zero:

n i=1 c i e i = c 1 . . . c n    e 1 . . . e n    = c 1 . . . c n X    v 1 . . . v m    = 0

contradicting the fact that e i are linearly independent. Hence we conclude the theorem for r = 1.

As for r ≥ 2, the image ρ img (A) is necessarily of the form i V ⊕d i i (by Corollary 20. 1.3 ) and by the above

d i = dim V i for each i.

Example 20.2.3 (Applying the proof of density theorem on an explicit example)

We can run through the argument on an explicit example to better understand how it works -in order to do this, we need V to be an irrep, otherwise the image of Reg(A) would not be isomorphic to V ⊕m , and we will not be able to run to the end of the argument.

Let A = Mat 2 (k), and V ∼ = k ⊕2 with the obvious action. As we know, this is an irrep.

The density theorem claims that ρ : A → Mat(V ) is surjective, which means for any e 1 , e 2 ∈ V independent, and any

w 1 , w 2 ∈ V , we can find a ∈ A such that a • (e 1 , e 2 ) = (w 1 , w 2 ).

Because we're working through a counterexample, pick e 1 = (1, 0), e 2 = (2, 0) instead. Then, for some w 1 , w 2 ∈ V , there may be no a that sends e 1 to w 1 to e 2 to w 2 .

Consider the representation morphism Reg

(A) → V ⊕2 by a → (a • e 1 , a • e 2 ); its image is thus {(v, 2v) | v ∈ V }, which is a subrepresentation of V ⊕2 , isomorphic as a representation to V ⊕1 ∼ = V by v → (v, 2v) = v 1 2 . Then, we can find v = 1 0 ∈ V ⊕1 , for which e 1 e 2 = v 1 2 .

Now, with the explicit array of numbers 1 2 , it is easy to find a linear dependence on e 1 and e 2 . §20.3 Semisimple algebras

Definition 20.3.1. A finite-dimensional algebra A is a semisimple if every finite- dimensional representation of A is completely reducible.

Theorem 20.

3.2 (Semisimple algebras)

Let A be a finite-dimensional algebra. Then the following are equivalent:

(i) A ∼ = i Mat d i (k) for some d i .

(ii) A is semisimple.

(iii) Reg(A) is completely reducible. On one hand, by Problem 19D ⋆ , it is isomorphic to A op (A with opposite multiplication), because the only intertwining operators Reg(A) → Reg(A) are those of the form -• a.

On the other hand, suppose that we have set Reg

(A) = i V ⊕n i i

. By Theorem 20. 1 .2, we have

A op ∼ = Hom rep (Reg(A), Reg(A)) = i Mat n i ×n i (k).

But Mat n (k) op ∼ = Mat n (k) (just by transposing), so we recover the desired conclusion.

Remark 20.3.3 -The trick of the proof above resembles Cayley's theorem (Problem 1F † ), in that we make the object act on itself to get an explicit representation.

Remark 20.3.4 -We can compare this to Corollary 18.3.2. Here, any finitedimensional representation of A is a finite-dimensional left A-module, and from the theorem above, we know that if A is semisimple, any such module can be broken down into a direct sum of irreps

V i ∼ = k ⊕d i .

Note that unlike the case where A is a PID, k ⊕d i is not isomorphic to a quotient of the ring Mat d i (k).

In fact, if we combine the above result with the density theorem (and Corollary 20.2.2), we obtain: Theorem 20. 3 

.5 (Sum of squares formula)

For a finite-dimensional algebra A we have

i dim(V i ) 2 ≤ dim A

where the V i are the irreps of A; equality holds exactly when A is semisimple, in which case Reg(A)

∼ = i Mat(V i ) ∼ = I V ⊕ dim V i i .

Proof. 

A → i Mat(V i ) is an isomorphism; this means all V i are present.

Remark 20.3.6 (Digression) -For any finite-dimensional A, the kernel of the map ρ :

A → i Mat(V i ) is denoted Rad(A)

and is the so-called Jacobson radical of A; it's the set of all a ∈ A which act by zero in all irreps of A. The usual definition of "semisimple" given in books is that this Jacobson radical is trivial.

§20.4 Maschke's theorem

We now prove that the representation theory of groups is as nice as possible. This tells us that when studying representations of groups, all representations are completely reducible.

Proof. Consider any finite

-dimensional representation (V, ρ) of k[G]. Given a proper subrepresentation W ⊆ V , our goal is to construct a supplementary G-invariant subspace W ′ which satisfies V = W ⊕ W ′ . This will show that indecomposable ⇐⇒ irreducible, which is enough to show k[G] is semisimple. Let π : V → W be any projection of V onto W , meaning π(v) = v ⇐⇒ v ∈ W . We consider the averaging map P : V → V by P = 1 |G| g∈G ρ(g -1 ) • π • ρ(g).

We'll use the following properties of the map:

Exercise 20.4.2. Show that the map P satisfies:

• For any w ∈ W , P (w) = w.

• For any v ∈ V , P (v) ∈ W .

• The map P : V → V is an intertwining operator.

Thus P is idempotent (it is the identity on its image W ), so by Problem 9G ⋆ we have V = ker P ⊕ im P , but both ker P and im P are subrepresentations as desired.

Remark 20.4.3 -In the case where k = C, there is a shorter proof. Suppose B : V × V → C is an arbitrary bilinear form. Then we can "average" it to obtain a new bilinear form ⟨v, w⟩

:= 1 |G| g∈G B(g • v, g • w). The averaged form ⟨-, -⟩ is G-invariant, in the sense that ⟨v, w⟩ = ⟨g • v, g • w⟩.

Then, one sees that if W ⊆ V is a subrepresentation, so is its orthogonal complement W ⊥ . This implies the result.

§20.5 Example: the representations of C[S 3 ]

We compute all irreps of C[S 3 ]. I'll take for granted right now there are exactly three such representations (which will be immediate by the first theorem in the next chapter: we'll in fact see that the number of representations of G is exactly equal to the number of conjugacy classes of G).

Given that, if the three representations of have dimension

d 1 , d 2 , d 3 , then we ought to have d 2 1 + d 2 2 + d 2 3 = |G| = 6.

From this, combined with some deep arithmetic, we deduce that we should have d 1 = d 2 = 1 and d 3 = 2 or some permutation.

In fact, we can describe these representations explicitly. First, we define: Another way to see this is that the trivial representation depends on how the k-algebra is written as a group algebra: k[Z/2Z] has a k-algebra automorphism given by g → -g, where g is the generator of the group Z/2Z; however the corresponding trivial representations are different.

Then the representations are:

• The one-dimensional C triv ; each σ ∈ S 3 acts by the identity.

• There is a nontrivial one-dimensional representation C sign where the map S 3 → C × is given by sending σ to the sign of σ. Thus in C sign every σ ∈ S 3 acts as ±1. Of course, C triv and C sign are not isomorphic (as one-dimensional representations are never isomorphic unless the constants they act on coincide for all a, as we saw in Problem 19A † ).

• Finally, we have already seen the two-dimensional representation, but now we give it a name. Define refl 0 to be the representation whose vector space is {(x, y, z) | x + y + z = 0}, and whose action of S 3 on it is permutation of coordinates.

Exercise 20.5.3. Show that refl 0 is irreducible, for example by showing directly that no subspace is invariant under the action of S 3 .

Thus V is also not isomorphic to the previous two representations.

This implies that these are all the irreps of S 3 . Note that, if we take the representation V of S 3 on k ⊕3 , we just get that V = refl 0 ⊕C triv . §20.6 A few harder problems to think about

Problem 20A. Find all the irreps of C[Z/nZ].

Problem 20B (Maschke requires |G| finite). Consider the representation of the group R on C ⊕2 under addition by a homomorphism

R → Mat 2 (C) by t → 1 t 0 1 .

Show that this representation is not irreducible, but it is indecomposable.

Problem 20C.

Prove that all irreducible representations of a finite group are finitedimensional.

Problem 20D. Determine all the complex irreps of D 10 .

Problem 20E (AIME 2018). The wheel shown below consists of two circles and five spokes, with a label where a spoke meets a circle. A bug walks along the wheel, starting from A. The bug takes 15 steps. At each step, the bug moves to an adjacent label such that it only walks counterclockwise along the inner circle and clockwise along the outer circle. In how many ways can the bug move to end up at A after all steps?

A Characters are basically the best thing ever. To every representation V of A we will attach a so-called character χ V : A → k. It will turn out that the characters of irreps of V will determine the representation V completely. Thus an irrep is just specified by a set of dim A numbers. §21.1 Definitions

Definition 21.1.1. Let V = (V, ρ) be a finite-dimensional representation of A. The character χ V : A → k attached to A is defined by χ V = Tr •ρ, i.e. χ V (a) := Tr (ρ(a) : V → V ) .

Since Tr and ρ are additive, this is a k-linear map (but it is not multiplicative). Note also that χ V ⊕W = χ V + χ W for any representations V and W .

We are especially interested in the case

A = k[G], of course.

As usual, we just have to specify χ V (g) for each g ∈ G to get the whole map k[G] → k. Thus we often think of χ V as a function G → k, called a character of the group G. Here is the case

G = S 3 : Example 21.1.2 (Character table of S 3 )

Let's consider the three irreps of G = S 3 from before. For C triv all traces are 1; for C sign the traces are ±1 depending on sign (obviously, for one-dimensional maps k → k the trace "is" just the map itself). For refl 0 we take a basis (1, 0, -1) and (0, 1, -1), say, and compute the traces directly in this basis.

χ V (g) id (1 2) (2 3) (3 1) (1 2 3) (3 2 1) C triv 1 1 1 1 1 1 C sign 1 -1 -1 -1 1 1 refl 0 2 0 0 0 -1 -1

The above table is called the character table of the group G. The table above has certain mysterious properties, which we will prove as the chapter progresses.

(I) The value of χ V (g) only depends on the conjugacy class of g.

(II) The number of rows equals the number of conjugacy classes.

(III) The sum of the squares of any row is 6 again! (IV) The "dot product" of any two rows is zero.

Abuse of Notation 21.1.3. The name "character" for χ V : G → k is a bit of a misnomer. This χ V is not multiplicative in any way, as the above example shows: one can almost think of it as an element of k ⊕|G| .

Question 21. 1.4 . Show that χ V (1 A ) = dim V , so one can read the dimensions of the representations from the leftmost column of a character table.

§21.2 The dual space modulo the commutator

For any algebra, we first observe that since Tr(T S) = Tr(ST ), we have for any V that

χ V (ab) = χ V (ba).

This explains observation (I) from earlier: When

A = k[G], the space A ab is not isomorphic to the group algebra k[G/[G, G]

]! This is because, in the abelianization of the group G, we identify ghg -1 h -1 = 1 for all g, h ∈ G, which is not the same as gh -hg.

In fact, in the general case, A ab does not even inherit the structure of a k-algebra from A, it can only get a k-vector space structure.

Theorem 21.2.5 (Character of representations of algebras)

Let A be an algebra over an algebraically closed field. Then (a) Characters of pairwise non-isomorphic irreps are linearly independent in (A ab ) ∨ .

(b) If A is finite-dimensional and semisimple, then the characters attached to irreps form a basis of (A ab ) ∨ .

In particular, in (b) the number of irreps of A equals dim A ab .

Proof. Part (a) is more or less obvious by the density theorem: suppose there is a linear dependence, so that for every a we have

c 1 χ V 1 (a) + c 2 χ V 2 (a) + • • • + c r χ Vr (a) = 0

for some integer r. For part (b), assume there are r irreps. We may assume that

A = r i=1 Mat(V i )

where V 1 , . . . , V r are the irreps of A. Since we have already showed the characters are linearly independent we need only show that dim(A/[A, A]) = r, which follows from the observation earlier that each Mat(V i ) has a one-dimensional abelianization.

Since G has dim k[G] ab conjugacy classes, this completes the proof of (II).

§21.3 Orthogonality of characters

Now we specialize to the case of finite groups G, represented over C. 

⟨f 1 , f 2 ⟩ = 1 |G| g∈G f 1 (g)f 2 (g).

This is the same "dot product" that we mentioned at the beginning, when we looked at the character table of S 3 . We now aim to prove the following orthogonality theorem, which will imply (III) and (IV) from earlier.

Theorem 21.3.4 (Orthogonality)

For any finite-dimensional complex representations V and W of G we have

⟨χ V , χ W ⟩ = dim Hom rep (W, V ).

In particular, if V and W are irreps then

⟨χ V , χ W ⟩ = 1 V ∼ = W 0 otherwise.

Corollary 21.3.5 (Irreps give an orthonormal basis)

The characters associated to irreps form an orthonormal basis of Fun class (G).

In order to prove this theorem, we have to define the dual representation and the tensor representation, which give a natural way to deal with the quantity χ V (g)χ W (g).

Definition 21.3.6. Let V = (V, ρ) be a representation of G. The dual representation V ∨ is the representation on V ∨ with the action of G given as follows: for each ξ ∈ V ∨ , the action of g gives a g • ξ ∈ V ∨ specified by

v g•ξ --→ ξ ρ(g -1 )(v) . Definition 21.3.7. Let V = (V, ρ V ) and W = (W, ρ W ) be group representations of G.

The tensor product of V and W is the group representation on V ⊗ W with the action of G given on pure tensors by

g • (v ⊗ w) = (ρ V (g)(v)) ⊗ (ρ W (g)(w))

which extends linearly to define the action of G on all of V ⊗ W .

Remark 21.3.8 -Warning: the definition for tensors does not extend to algebras. We might hope that a

• (v ⊗ w) = (a • v) ⊗ (a • w) would work, but this is not even linear in a ∈ A (

what happens if we take a = 2, for example?).

Theorem 21.3.9 (Character traces)

If V and W are finite-dimensional representations of G, then for any g ∈ G. 

(a) χ V ⊕W (g) = χ V (g) + χ W (g). (b) χ V ⊗W (g) = χ V (g) • χ W (g). (c) χ V ∨ (g) = χ V (g).

(ρ(g)) |G| = ρ(g |G| ) = ρ(1 G ) = id V

by Lagrange's theorem, we can diagonalize ρ(g), say with eigenvalues λ 1 , . . . , λ n which are |G|th roots of unity, corresponding to eigenvectors e 1 , . . . , e n . Then we see that in the basis e ∨ 1 , . . . , e ∨ n , the action of g on

V ∨ has eigenvalues λ -1 1 , λ -1 2 , . . . , λ -1 n . So χ V (g) = n i=1 λ i and χ V ∨ (g) = n i=1 λ -1 i = n i=1 λ i

where the last step follows from the identity |z| = 1 ⇐⇒ z -1 = z.

Remark 21.3.10 (Warning) -The identities (b) and (c) do not extend linearly to

C[G], i.e. it is not true for example that χ V ∨ (a) = χ V (a) if we think of χ V as a map C[G] → C.

Proof of orthogonality relation. The key point is that we can now reduce the sums of products to just a single character by

χ V (g)χ W (g) = χ V ⊗W ∨ (g).

So we can rewrite the sum in question as just

⟨χ V , χ W ⟩ = 1 |G| g∈G χ V ⊗W ∨ (g) = χ V ⊗W ∨   1 |G| g∈G g   .

Let Hence V ⊗ W ∨ = ker P ⊕ im P (by Problem 9G ⋆ ) and im P is the subspace of elements which are fixed under G. From this we deduce that

P : V ⊗ W ∨ → V ⊗ W ∨ be the action of

Tr P = dim im P = dim x ∈ V ⊗ W ∨ | g • x = x ∀g ∈ G . Now, consider the natural isomorphism V ⊗ W ∨ → Hom(W, V ). Exercise 21.3.12. Let g ∈ G. Show that under this isomorphism, T ∈ Hom(W, V ) satisfies g • T = T if and only if T (g • w) = g • T (w) for each w ∈ W .

(This is just unwinding three or four definitions.)

Consequently, χ V ⊗W ∨ (P ) = Tr P = dim Hom rep (W, V ) as desired.

The orthogonality relation gives us a fast and mechanical way to check whether a finite-dimensional representation V is irreducible. Namely, compute the traces χ V (g) for each g ∈ G, and then check whether ⟨χ V , χ V ⟩ = 1. So, for example, we could have seen the three representations of S 3 that we found were irreps directly from the character table. Thus, we can now efficiently verify any time we have a complete set of irreps.

§21.4 Examples of character tables

Example 21.4.1 (Dihedral group on 10 elements)

Let D 10 = r, s | r 5 = s 2 = 1, rs = sr -1 . Let ω = exp( 2πi 5

). We write four representations of D 10 :

• C triv , all elements of D 10 act as the identity.

• C sign , r acts as the identity while s acts by negation.

• V 1 , which is two-dimensional and given by r → ω 0 0 ω 4 and s → 0 1 1 0 .

• V 2 , which is two-dimensional and given by r → ω 2 0 0 ω 3 and s → 0 1 1 0 .

We claim that these four representations are irreducible and pairwise non-isomorphic.

We do so by writing the character table:

D 10 1 r, r 4 r 2 , r 3 sr k C triv 1 1 1 1 C sign 1 1 1 -1 V 1 2 ω + ω 4 ω 2 + ω 3 0 V 2 2 ω 2 + ω 3 ω + ω 4 0

Then a direct computation shows the orthogonality relations, hence we indeed have an orthonormal basis. For example, ⟨C triv ,

C sign ⟩ = 1 + 2 • 1 + 2 • 1 + 5 • (-1) = 0. Example 21.4.2 (Character table of S 4 )

We now have enough machinery to compute the character table of S 4 , which has five conjugacy classes (corresponding to cycle types id, 2, 3, 4 and 2 + 2). First of all, we note that it has two one-dimensional representations, C triv and C sign , and these are the only ones (because there are only two homomorphisms S 4 → C × ). So thus far we have the table

S 4 1 (• •) (• • •) (• • • •) (• •)(• •) C triv 1 1 1 1 1 C sign 1 -1 1 -Note the columns represent 1 + 6 + 8 + 6 + 3 = 24 elements. Now, the remaining three representations have dimensions

d 1 , d 2 , d 3 with d 2 1 + d 2 2 + d 2 3 = 4! -2 = 22

which has only (d 1 , d 2 , d 3 ) = (2, 3, 3) and permutations. Now, we can take the refl 0 representation {(w, x, y, z)

| w + x + y + z = 0}

with basis (1, 0, 0, -1), (0, 1, 0, -1) and (0, 0, 1, -1). This can be geometrically checked to be irreducible, but we can also do this numerically by computing the character directly (this is tedious): it comes out to have 3, 1, 0, -1, -1 which indeed gives norm

⟨χ refl 0 , χ refl 0 ⟩ = 1 4!    3 2 id + 6 • (1) 2 (• •) + 8 • (0) 2 (• • •) + 6 • (-1) 2 (• • • •) + 3 • (-1) 2 (• •)(• •)    = 1.

Note that we can also tensor this with the sign representation, to get another irreducible representation (since C sign has all traces ±1, the norm doesn't change). Finally, we recover the final row using orthogonality (which we name C 2 , for lack of a better name); hence the completed table is as follows.

S 4 1 (• •) (• • •) (• • • •) (• •)(• •) C triv 1 1 1 1 1 C sign 1 -1 1 -1 1 C 2 2 0 -1 0 2 refl 0 3 1 0 -1 -1 refl 0 ⊗C sign 3 -1 0 1 -1 §21.

5 A few harder problems to think about

Problem 21A † (Reading decompositions from characters). Let W be a complex representation of a finite group G. Let V 1 , . . . , V r be the complex irreps of G and set

n i = ⟨χ W , χ V i ⟩.

Prove that each n i is a non-negative integer and

W = r i=1 V ⊕n i i .

Problem 21B. Consider complex representations of G = S 4 . The representation refl 0 ⊗ refl 0 is 9-dimensional, so it is clearly reducible. Compute its decomposition in terms of the five irreducible representations.

Problem 21C (Tensoring by one-dimensional irreps). Let V and W be irreps of G, with dim

W = 1. Show that V ⊗ W is irreducible.

Problem 21D (Quaternions). Compute the character table of the quaternion group Q 8 .

Problem 21E ⋆ (Second orthogonality formula). Let g and h be elements of a finite group G, and let V 1 , . . . , V r be the irreps of G. Prove that

r i=1 χ V i (g)χ V i (h) = |C G (g)| if g and h are conjugates 0 otherwise.

Here,

C G (g) = {x ∈ G : xg = gx} is the centralizer of g.

With all this setup, we now take the time to develop some nice results which are of independent interest. §22.1 Frobenius divisibility Theorem 22.1.1 (Frobenius divisibility)

Let V be a complex irrep of a finite group G. Then dim V divides |G|.

The proof of this will require algebraic integers (developed in the algebraic number theory chapter). Recall that an algebraic integer is a complex number which is the root of a monic polynomial with integer coefficients, and that these algebraic integers form a ring Z under addition and multiplication, and that Z ∩ Q = Z.

First, we prove:

Lemma 22.1.2 (Elements of Z[G] are integral) Let α ∈ Z[G].

Then there exists a monic polynomial P with integer coefficients such that P (α) = 0. . . cannot all be strict, hence A k = A k+1 for some k, which means α k+1 can be expressed in terms of lower powers of α.

Proof. Let A k be the Z-span of 1, α 1 , . . . , α k . Since Z[G] is Noetherian, the inclusions A 0 ⊆ A 1 ⊆ A 2 ⊆ .

Proof of Frobenius divisibility. Let C 1 , . . . , C m denote the conjugacy classes of G. Then consider the rational number |G| dim V ;

we will show it is an algebraic integer, which will prove the theorem. Observe that we can rewrite it as

|G| dim V = |G| ⟨χ V , χ V ⟩ dim V = g∈G χ V (g)χ V (g) dim V .

We split the sum over conjugacy classes, so

|G| dim V = m i=1 χ V (C i ) • |C i |χ V (C i ) dim V .

We claim that for every i,

|C i |χ V (C i ) dim V = 1 dim V Tr T i

is an algebraic integer, where

T i := ρ   h∈C i h   .

To see this, note that T i commutes with elements of G, and hence is an intertwining operator T i : V → V . Thus by Schur's lemma, T i = λ i • id V and Tr T = λ i dim V . By Lemma 22. 1.2 , λ i ∈ Z, as desired. Now we are done, since χ V (C i ) ∈ Z too (it is the sum of conjugates of roots of unity), so |G| dim V is the sum of products of algebraic integers, hence itself an algebraic integer.

§22.2 Burnside's theorem

We now prove a group-theoretic result. This is the famous poster child for representation theory (in the same way that RSA is the poster child of number theory) because the result is purely group theoretic.

Recall that a group is simple if it has no normal subgroups. In fact, we will prove:

Theorem 22.2.1 (Burnside)

Let G be a nonabelian group of order p a q b (where p, q are distinct primes and a, b ≥ 0). Then G is not simple.

In what follows p and q will always denote prime numbers.

Lemma 22.2.2 (On gcd(|C|, dim V ) = 1) Let V = (V, ρ) be an complex irrep of G. Assume C is a conjugacy class of G with gcd(|C|, dim V ) = 1. Then for any g ∈ C, either • ρ(g) is multiplication by a scalar, or • χ V (g) = Tr ρ(g) = 0.

Proof. If ε i are the n eigenvalues of ρ(g) (which are roots of unity), then from the proof of Frobenius divisibility we know

|C| n χ V (g) ∈ Z, thus from gcd(|C|, n) = 1 we get 1 n χ V (g) = 1 n (ε 1 + • • • + ε n ) ∈ Z.

So this follows readily from a fact from algebraic number theory, namely Problem 47C ⋆ : either

ε 1 = • • • = ε n (first case) or ε 1 + • • • + ε n = 0 (second case).

Lemma

22.2.3 (Simple groups don't have prime power conjugacy classes)

Let G be a finite simple group. Then G cannot have a conjugacy class of order p k (where k > 0).

Proof. By contradiction. Assume C is such a conjugacy class, and fix any g ∈ C. By the second orthogonality formula (Problem 21E ⋆ ) applied g and 1 G (which are not conjugate since g ̸ = 1 G ) we have

r i=1 dim V i χ V i (g) = 0

where V i are as usual all irreps of G.

Exercise 22.2.4. Show that there exists a nontrivial irrep V such that p ∤ dim V and χ V (g) ̸ = 0. (Proceed by contradiction to show that -1 p ∈ Z if not.)

Let V = (V, ρ) be the irrep mentioned. By the previous lemma, we now know that ρ(g) acts as a scalar in V . Now consider the subgroup

H = ab -1 | a, b ∈ C ⊆ G.

We claim this is a nontrivial normal subgroup of G. It is easy to check H is normal, and since |C| > 1 we have that H is nontrivial. As represented by V each element of H acts trivially in G, so since V is nontrivial and irreducible, H ̸ = G. This contradicts the assumption that G was simple.

With this lemma, Burnside's theorem follows by partitioning the |G| elements of our group into conjugacy classes. Assume for contradiction G is simple. Each conjugacy class must have order either 1 (of which there are |Z(G)| by Problem 16D ⋆ ) or divisible by pq (by the previous lemma), but on the other hand the sum equals |G| = p a q b . Consequently, we must have

|Z(G)| > 1. But G is not abelian, hence Z(G) ̸ = G, thus the center Z(G)

is a nontrivial normal subgroup, contradicting the assumption that G was simple.

§22.3 Frobenius determinant

We finish with the following result, the problem that started the branch of representation theory. Given a finite group G, we create n variables {x g } g∈G , and an n × n matrix M G whose (g, h)th entry is x gh . 

M G = x id x T x T x id .

Then det

M G = (x id -x T )(x id + x T ). (b) If G = S 3 , a long computation gives the irreducible factorization of det M G is   σ∈S 3 x σ     σ∈S 3 sign(σ)x σ   F x id , x (123) , x (321) -F x (12) , x (23) , x (31) 2 where F (a, b, c) = a 2 + b 2 + c 2 -ab -bc -ca;

the latter factor is irreducible.

Theorem 22.3.2 (Frobenius determinant)

The polynomial det M G (in |G| variables) factors into a product of irreducible polynomials such that (i) The number of polynomials equals the number of conjugacy classes of G, and

(ii) The multiplicity of each polynomial equals its degree.

You may already be able to guess how the "sum of squares" result is related! (Indeed, look at deg det M G .) Legend has it that Dedekind observed this behavior first in 1896. He didn't know how to prove it in general, so he sent it in a letter to Frobenius, who created representation theory to solve the problem.

With all the tools we've built, it is now fairly straightforward to prove the result.

Proof. Let V = (V, ρ) = Reg(C[G]) and let V 1 , . . . , V r be the irreps of G. Let's consider the map T : C[G] → C[G] which has matrix M G in the usual basis of C[G],

namely

T : T ({x g } g∈G ) = g∈G x g ρ(g) ∈ Mat(V ).

Thus we want to examine det T .

But we know that

V = r i=1 V ⊕ dim V i i

as before, and so breaking down T over its subspaces we know

det T = r i=1 (det(T ↾ V i )) dim V i .

So we only have to show two things: the polynomials det T V i are irreducible, and they are pairwise different for different i.

Let V i = (V i , ρ), and pick k = dim V i .

• Irreducible: By the density theorem, for any M ∈ Mat(V i ) there exists a particular choice of complex numbers x g ∈ G such that

M = g∈G x g • ρ i (g) = (T ↾ V i )({x g }).

View ρ i (g) as a k × k matrix with complex coefficients. Thus the "generic" (T ↾ V i )({x g }), viewed as a matrix with polynomial entries, must have linearly independent entries (or there would be some matrix in Mat(V i ) that we can't achieve).

Then, the assertion follows (by a linear variable change) from the simple fact that the polynomial det(y ij ) 1≤i,j≤m in m 2 variables is always irreducible.

• Pairwise distinct: We show that from det T | V i ({x g }) we can read off the character χ V i , which proves the claim. In fact

Exercise 22.3.3. Pick any basis for V i . If dim V i = k, and 1 G ̸ = g ∈ G, then χ Vi (g) is the coefficient of x g x k-1 1 G .

Thus, we are done.

VII Quantum Algorithms

Part VII: Contents 

24

Quantum circuits 265 24.1 Classical logic gates . 

25

Shor's algorithm 273 25. 1 The classical (inverse) Fourier transform .

25.2 The quantum

In this chapter we'll explain how to set up quantum states using linear algebra. This will allow me to talk about quantum circuits in the next chapter, which will set the stage for Shor's algorithm.

I won't do very much physics (read: none at all). That is, I'll only state what the physical reality is in terms of linear algebras, and defer the philosophy of why this is true to your neighborhood "Philosophy of Quantum Mechanics" class (which is a "social science" class at MIT!).

§23.1 Bra-ket notation

Physicists have their own notation for vectors: whereas I previously used something like v, e 1 , and so on, in this chapter you'll see the infamous bra-ket notation: a vector will be denoted by |•⟩, where • is some variable name: unlike in math or Python, this can include numbers, symbols, Unicode characters, whatever you like. This is called a "ket". To pay a homage to physicists everywhere, we'll use this notation for this chapter too.

Abuse of Notation 23.1.1 (For this part, dim H < ∞).

In this part on quantum computation, we'll use the word "Hilbert space" as defined earlier, but in fact all our Hilbert spaces will be finite-dimensional. We even can write dot products succinctly in this notation: if |ϕ⟩ = γ δ , then the dot product of ⟨ψ| and |ϕ⟩ is given by

⟨ψ|ϕ⟩ = α β γ δ = αγ + βδ.

So we will use the notation ⟨ψ|ϕ⟩ instead of the more mathematical ⟨|ψ⟩ , |ϕ⟩⟩. In particular, the squared norm of |ψ⟩ is just ⟨ψ|ψ⟩. Concretely, for dim H = 2 we have

⟨ψ|ψ⟩ = |α| 2 + |β| 2 . §23.

2 The state space

If you think that's weird, well, it gets worse.

In classical computation, a bit is either 0 or 1. More generally, we can think of a classical space of n possible states 0, . . . , n -1. Thus in the classical situation, the space of possible states is just a discrete set with n elements.

In quantum computation, a qubit is instead any complex linear combination of 0 and 1. To be precise, consider the normed complex vector space

H = C ⊕2

and denote the orthonormal basis elements by |0⟩ and |1⟩. Then a qubit is a nonzero element |ψ⟩ ∈ H, so that it can be written in the form

|ψ⟩ = α |0⟩ + β |1⟩

where α and β are not both zero. Typically, we normalize so that |ψ⟩ has norm 1:

⟨ψ|ψ⟩ = 1 ⇐⇒ |α| 2 + |β| 2 = 1.

In particular, we can recover the "classical" situation with |0⟩ ∈ H and |1⟩ ∈ H, but now we have some "intermediate" states, such as

1 √ 2 (|0⟩ + |1⟩) .

Philosophically, what has happened is that:

Instead of allowing just the states |0⟩ and |1⟩, we allow any complex linear combination of them.

More generally, if dim H = n, then the possible states are nonzero elements

c 0 |0⟩ + c 1 |1⟩ + • • • + c n-1 |n -1⟩

which we usually normalize so that

|c 0 | 2 + |c 1 | 2 + • • • + |c n-1 | 2 = 1.

§23.3 Observations

Prototypical example for this section: id corresponds to not making a measurement since all its eigenvalues are equal, but any operator with distinct eigenvalues will cause collapse.

If you think that's weird, well, it gets worse. First, some linear algebra review (Definition 15.4.1): Definition 23.3.1. Let V be a finite-dimensional inner product space. For a map T : V → V , the following conditions are equivalent:

• ⟨T x, y⟩ = ⟨x, T y⟩ for any x, y ∈ V .

• T = T † .

A map T satisfying these conditions is called Hermitian. Thus, we know that T is diagonalizable with respect to the inner form, so for a suitable basis we can write it in an orthonormal basis as

T =       λ 0 0 . . . 0 0 λ 1 . . . 0 . . . . . . . . . . . . 0 0 . . . λ n-1      

.

As we've said, this is fantastic: not only do we have a basis of eigenvectors, but the eigenvectors are pairwise orthogonal, and so they form an orthonormal basis of V . Back to quantum computation. Suppose we have a state |ψ⟩ ∈ H, where dim H = 2; we haven't distinguished a particular basis yet, so we just have a nonzero vector. Then the way observations work (and this is physics, so you'll have to take my word for it) is as follows:

Pick a Hermitian operator T : H → H; then observations of T return eigenvalues of T .

To be precise:

• Pick a Hermitian operator T : H → H, which is called the observable.

• Consider its eigenvalues λ 0 , . . . , λ n-1 and corresponding eigenvectors |0⟩ T , . . . , |n -1⟩ T . Tacitly we may assume that |0⟩ T , . . . , |n -1⟩ T form an orthonormal basis of H. (The subscript T is here to distinguish the eigenvectors of T from the basis elements of H.)

• Write |ψ⟩ in the orthonormal basis as

c 0 |0⟩ T + c 1 |1⟩ T + • • • + c n-1 |n -1⟩ T .

• Then the probability of observing λ i is

|c i | 2 |c 0 | 2 + • • • + |c n-1 | 2 .

This is called making an observation along T .

Note that in particular, for any nonzero constant c, |ψ⟩ and c |ψ⟩ are indistinguishable, which is why we like to normalize |ψ⟩. But the queerest thing of all is what happens to |ψ⟩: by measuring it, we actually destroy information. This behavior is called quantum collapse.

• Suppose for simplicity that we observe |ψ⟩ with T and obtain an eigenvalue λ, and that |i⟩ T is the only eigenvector with this eigenvalue. Then, the state |ψ⟩ collapses to just the state c i |i⟩ T : all the other information is destroyed. ( In fact, we may as well say it collapses to |i⟩ T , since again constant factors are not relevant.)

• More generally, if we observe λ, consider the generalized eigenspace H λ (i.e. the span of eigenvectors with the same eigenvalue). Then the physical state |ψ⟩ has been changed as well: it has now been projected onto the eigenspace H λ . In still other words, after observation, the state collapses to

0≤i≤n λ i =λ c i |i⟩ T .

In other words,

When we make a measurement, the coefficients from different eigenspaces are destroyed.

Why does this happen? Beats me. 

|ψ⟩ = i √ 5 |0⟩ + 2 √ 5 |1⟩ = i/ √ 5 2/ √ 5 ∈ H. (a) Let T = 1 0 0 -1 .

This has eigenvectors |0⟩ = |0⟩ T and |1⟩ = |1⟩ T , with eigenvalues +1 and -1. So if we measure |ψ⟩ to T , we get +1 with probability 1/5 and -1 with probability 4/5. After this measurement, the original state collapses to |0⟩ if we measured +1, and |1⟩ if we measured -1. So we never learn the original probabilities.

(b) Now consider T = id, and arbitrarily pick two orthonormal eigenvectors

|0⟩ T , |1⟩ T ; thus ψ = c 0 |0⟩ T + c 1 |1⟩ T .

Since all eigenvalues of T are +1, our measurement will always be +1 no matter what we do. But there is also no collapsing, because none of the coefficients get destroyed.

(c) Now consider

T = 0 7 7 0 .

The two normalized eigenvectors are

|0⟩ T = 1 √ 2 1 1 |1⟩ T = 1 √ 2 1 -1

with eigenvalues +7 and -7 respectively. In this basis, we have

|ψ⟩ = 2 + i √ 10 |0⟩ T + -2 + i √ 10 |1⟩ T .

So we get +7 with probability For H = C ⊕2 we can come up with more classes of examples using the so-called Pauli matrices. These are the three Hermitian matrices

σ z = 1 0 0 -1 σ x = 0 1 1 0 σ y = 0 -i i 0 .

These matrices are important because:

Question 23.3.6. Show that these three matrices, plus the identity matrix, form a basis for the set of Hermitian 2 × 2 matrices.

So the Pauli matrices are a natural choice of basis. 1Their normalized eigenvectors are

|↑⟩ = |0⟩ = 1 0 |↓⟩ = |1⟩ = 0 1 |→⟩ = 1 √ 2 1 1 |←⟩ = 1 √ 2 1 -1 |⊗⟩ = 1 √ 2 1 i |⊙⟩ = 1 √ 2 1 -i

which we call "z-up", "z-down", "x-up", "x-down", "y-up", "y-down" respectively. (The eigenvalues are +1 for "up" and -1 for "down".) So, given a state |ψ⟩ ∈ C ⊕2 we can make a measurement with respect to any of these three bases by using the corresponding Pauli matrix.

In light of this, the previous examples were (a) measuring along σ z , (b) measuring along id, and (c) measuring along 7σ x .

Notice that if we are given a state |ψ⟩, and are told in advance that it is either |→⟩ or |←⟩ (or any other orthogonal states) then we are in what is more or less a classical situation. Specifically, if we make a measurement along σ x , then we find out which state that |ψ⟩ was in (with 100% certainty), and the state does not undergo any collapse. Thus, orthogonal states are reliably distinguishable. §23.4 Entanglement Prototypical example for this section: Singlet state: spooky action at a distance.

If you think that's weird, well, it gets worse. Qubits don't just act independently: they can talk to each other by means of a tensor product. Explicitly, consider

H = C ⊕2 ⊗ C ⊕2

endowed with the norm described in Problem 13D ⋆ . One should think of this as a qubit A in a space H A along with a second qubit B in a different space H B , which have been allowed to interact in some way, and H = H A ⊗ H B is the set of possible states of both qubits. Thus

|0⟩ A ⊗ |0⟩ B , |0⟩ A ⊗ |1⟩ B , |1⟩ A ⊗ |0⟩ B , |1⟩ A ⊗ |1⟩ B

is an orthonormal basis of H; here |i⟩ A is the basis of the first C ⊕2 while |i⟩ B is the basis of the second C ⊕2 , so these vectors should be thought of as "unrelated" just as with any tensor product. The pure tensors mean exactly what you want: for example |0⟩ A ⊗ |1⟩ B means "0 for qubit A and 1 for qubit B".

As before, a measurement of a state in H requires a Hermitian map H → H. In particular, if we only want to measure the qubit B along M B , we can use the operator

id A ⊗ M B .

The eigenvalues of this operator coincide with the ones for M B , and the eigenspace for λ will be the H A ⊗ (H B ) λ , so when we take the projection the A qubit will be unaffected.

This does what you would hope for pure tensors in H:

Example 23.4.1 (Two non-entangled qubits)

Suppose we have qubit A in the state i

√ 5 |0⟩ A + 2 √ 5 |1⟩ A and qubit B in the state 1 √ 2 |0⟩ B + 1 √ 2 |1⟩ B .

So, the two qubits in tandem are represented by the pure tensor

|ψ⟩ = i √ 5 |0⟩ A + 2 √ 5 |1⟩ A ⊗ 1 √ 2 |0⟩ B + 1 √ 2 |1⟩ B .

Suppose we measure |ψ⟩ along

M = id A ⊗ σ B z .

The eigenspace decomposition is 

+ 2 √ 10 2 = 1 2 .

and the second eigenspace with probability 1 2 as well. (Note how the coefficients for A don't do anything!) After the measurement, we destroy the coefficients of the other eigenspace; thus (after re-normalization) we obtain the collapsed state

i √ 5 |0⟩ A + 2 √ 5 |1⟩ A ⊗ |0⟩ B or i √ 5 |0⟩ A + 2 √ 5 |1⟩ A ⊗ |1⟩ B

again with 50% probability each.

So this model lets us more or less work with the two qubits independently: when we make the measurement, we just make sure to not touch the other qubit (which corresponds to the identity operator). 

= C ⊕2 ⊗ C ⊕2 , say |ψ⟩ = α |00⟩ + β |01⟩ + γ |10⟩ + δ |11⟩ .

We can make a measurement along the diagonal matrix T : H → H with

T (|00⟩) = 0 |00⟩ , T (|01⟩) = 1 |01⟩ , T (|10⟩) = 2 |10⟩ , T (|11⟩) = 3 |11⟩ .

Thus we get each of the eigenvalues 0, 1, 2, 3 with probability |α| 2 , |β| 2 , |γ| 2 , |δ| 2 . So if we like we can make "simultaneous" measurements on two qubits in the same way that we make measurements on one qubit.

However, some states behave very weirdly.

Example 23.4.5 (The singlet state)

Consider the state

|Ψ -⟩ = 1 √ 2 |01⟩ - 1 √ 2 |10⟩

which is called the singlet state. 

T = σ A z ⊗ id B .

The eigenspace decomposition of T can be described as:

• The span of |00⟩ and |01⟩, with eigenvalue +1.

• The span of |10⟩ and |11⟩, with eigenvalue -1.

So one of two things will happen:

• With probability 1 2 , we measure +1 and the collapsed state is |01⟩.

• With probability 1 2 , we measure -1 and the collapsed state is |10⟩.

But now we see that measurement along A has told us what the state of the bit B is completely!

By solely looking at measurements on A, we learn B; this paradox is called spooky action at a distance, or in Einstein's tongue, spukhafte Fernwirkung. Thus,

In tensor products of Hilbert spaces, states which are not pure tensors correspond to "entangled" states.

What this really means is that the qubits cannot be described independently; the state of the system must be given as a whole. That's what entangled states mean: the qubits somehow depend on each other. §23.5 A few harder problems to think about Problem 23A. We measure |Ψ -⟩ by σ A

x ⊗ id B , and hence obtain either +1 or -1. Determine the state of qubit B from this measurement.

Problem 23B (Greenberger-Horne-Zeilinger paradox). Consider the state in (C

⊕2 ) ⊗3 |Ψ⟩ GHZ = 1 √ 2 (|0⟩ A |0⟩ B |0⟩ C -|1⟩ A |1⟩ B |1⟩ C ) .

Find the value of the measurements along each of

σ A y ⊗ σ B y ⊗ σ C x , σ A y ⊗ σ B x ⊗ σ C y , σ A x ⊗ σ B y ⊗ σ C y , σ A x ⊗ σ B x ⊗ σ C x .

As for the paradox: what happens if you multiply all these measurements together?

Now that we've discussed qubits, we can talk about how to use them in circuits. The key change -and the reason that quantum circuits can do things that classical circuits cannot -is the fact that we are allowing linear combinations of 0 and 1.

§24.1 Classical logic gates

In classical logic, we build circuits which take in some bits for input, and output some more bits for input. These circuits are built out of individual logic gates. For example, the AND gate can be pictured as follows. One can also represent the AND gate using the "truth table ":

A B A and B 0 0 0 0 1 0 1 0 0 1 1 1

Similarly, we have the OR gate and the NOT gate:

A B A or B 0 0 0 0 1 1 1 0 1 1 1 1 A not A 0 1 1 0

We also have a so-called COPY gate, which duplicates a bit.

0 copy 0 0 1 copy 1 1

Of course, the first theorem you learn about these gates is that: Theorem 24. 1 

.1 (AND, OR, NOT, COPY are universal)

The set of four gates AND, OR, NOT, COPY is universal in the sense that any boolean function f : {0, 1} n → {0, 1} can be implemented as a circuit using only these gates.

Proof. Somewhat silly: we essentially write down a circuit that OR's across all input strings in f pre (1). For example, suppose we have n = 3 and want to simulate the function f (abc) with f (011) = f (110) = 1 and 0 otherwise. Then the corresponding Boolean expression for f is simply Clearly, one can do the same for any other f , and implement this logic into a circuit.

f (abc) = [(not

Remark 24.1.2 -Since x and y = not ((not x) or (not y)), it follows that in fact, we can dispense with the AND gate.

§24.2 Reversible classical logic

Prototypical example for this section: CNOT gate, Toffoli gate.

For the purposes of quantum mechanics, this is not enough. To carry through the analogy we in fact need gates that are reversible, meaning the gates are bijections from the input space to the output space. In particular, such gates must take the same number of input and output gates. (b) The NOT gate, however, is reversible: it is a bijection {0, 1} → {0, 1}.

Example 24.2.2 (The CNOT gate)

The controlled-NOT gate, or the CNOT gate, is a reversible 2-bit gate with the following truth table.

In Out 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0

In other words, this gate XOR's the first bit to the second bit, while leaving the first bit unchanged. It is depicted as follows.

x

• x y x + y mod 2

The first dot is called the "control", while the ⊕ is the "negation" operation: the first bit controls whether the second bit gets flipped or not. Thus, a typical application might be as follows. • As input, x 1 . . . x n plus some fixed bits set to 0 or 1, called ancilla bits 1 .

• As output, the input bits x 1 , . . . , x n , the output bit f (x 1 , . . . , x n ), and possibly some extra bits (called garbage bits). 1 The English word "ancilla" means "maid".

The gate(s) are universal if they can simulate any Boolean function.

For example, the CNOT gate can simulate the NOT gate, using a single ancilla bit 1, according to the following circuit.

x • x 1 not x Unfortunately, it is not universal.

Proposition 24.2.4 (CNOT ̸ ⇒ AND)

The CNOT gate cannot simulate the boolean function "x and y". Proof. One can see that any function simulated using only CNOT gates must be of the form

Sketch of

a 1 x 1 + a 2 x 2 + • • • + a n x n (mod 2)

because CNOT is the map (x, y) → (x, x + y). Thus, even with ancilla bits, we can only create functions of the form ax + by + c (mod 2) for fixed a, b, c. The AND gate is not of this form.

So, we need at least a three-qubit gate. The most commonly used one is:

Definition 24.2.5. The three-bit Toffoli gate, also called the CCNOT gate, is given by

x • x y • y z z + xy (mod 2)

So the Toffoli has two controls, and toggles the last bit if and only if both of the control bits are 1.

This replacement is sufficient.

Theorem 24.2.6 (Toffoli gate is universal)

The Toffoli gate is universal.

Proof. We will show it can reversibly simulate AND, NOT, hence OR, which we know is enough to show universality. (We don't need COPY because of reversibility.) For the AND gate, we draw the circuit

x • x y • y 0

x and y with one ancilla bit, and no garbage bits.

For the NOT gate, we use two ancilla 1 bits and one garbage bit:

1 • 1 z • z 1 not z

This completes the proof.

Hence, in theory we can create any classical circuit we desire using the Toffoli gate alone. Of course, this could require exponentially many gates for even the simplest of functions. Fortunately, this is NO BIG DEAL because I'm a math major, and having 2 n gates is a problem best left for the CS majors.

§24.3 Quantum logic gates

In quantum mechanics, since we can have linear combinations of basis elements, our logic gates will instead consist of linear maps. Moreover, in quantum computation, gates are always reversible, which was why we took the time in the previous section to show that we can still simulate any function when restricted to reversible gates (e.g. using the Toffoli gate).

First, some linear algebra:

Definition 24.3.1. Let V be a finite dimensional inner product space. Then for a map U : V → V , the following are equivalent:

• ⟨U (x), U (y)⟩ = ⟨x, y⟩ for x, y ∈ V . • U † is the inverse of U . • ∥x∥ = ∥U (x)∥ for x ∈ V .

The map U is called unitary if it satisfies these equivalent conditions.

Then

Quantum logic gates are unitary matrices.

In particular, unlike the classical situation, quantum gates are always reversible (and hence they always take the same number of input and output bits).

For example, consider the CNOT gate. Its quantum analog should be a unitary map

U CNOT : H → H, where H = C ⊕2 ⊗ C ⊕2 ,

given on basis elements by

U CNOT (|00⟩) = |00⟩ , U CNOT (|01⟩) = |01⟩ U CNOT (|10⟩) = |11⟩ , U CNOT (|11⟩) = |10⟩ .

So pictorially, the quantum CNOT gate is given by

|0⟩ • |0⟩ |0⟩ |0⟩ |0⟩ • |0⟩ |1⟩ |1⟩ |1⟩ • |1⟩ |0⟩ |1⟩ |1⟩ • |1⟩ |1⟩ |0⟩

OK, so what? The whole point of quantum mechanics is that we allow linear qubits to be in linear combinations of |0⟩ and |1⟩, too, and this will produce interesting results. For example, let's take

|←⟩ = 1 √ 2 (|0⟩ -|1⟩

) and plug it into the top, with |1⟩ on the bottom, and see what happens:

U CNOT (|←⟩ ⊗ |1⟩) = U CNOT 1 √ 2 (|01⟩ -|11⟩) = 1 √ 2 (|01⟩ -|10⟩) = |Ψ -⟩

which is the fully entangled singlet state! Picture:

|←⟩ • |Ψ -⟩ |1⟩

Thus, when we input mixed states into our quantum gates, the outputs are often entangled states, even when the original inputs are not entangled.

Example 24.3.2 (More examples of quantum gates) (a) Every reversible classical gate that we encountered before has a quantum analog obtained in the same way as CNOT: by specifying the values on basis elements. For example, there is a quantum Tofolli gate which for example sends

|1⟩ • |1⟩ |1⟩ • |1⟩ |0⟩ |1⟩ (b)

The Hadamard gate on one qubit is a rotation given by

1 √ 2 1 √ 2 1 √ 2 -1 √ 2

. 

Thus

|0⟩ • |0⟩ |ψ⟩ U |ψ⟩ |1⟩ • |1⟩ |ψ⟩ U U |ψ⟩

Thus, U is applied when the controlling bit is 1, and CNOT is the special case U = σ x . As before, we get interesting behavior if the control is mixed.

And now, some more counterintuitive quantum behavior. Suppose we try to use CNOT as a copy, with truth table.

In Out 0 0 0 0

1 0 1 1 0 1 0 1 1 1 1 0

The point of this gate is to be used with a garbage 0 at the bottom to try and simulate a "copy" operation. So indeed, one can check that

|0⟩ U |0⟩ |0⟩ |0⟩ |1⟩ U |1⟩ |0⟩ |1⟩

Thus we can copy |0⟩ and |1⟩. But as we've already seen if we input |←⟩ ⊗ |0⟩ into U , we end up with the entangled state |Ψ -⟩ which is decisively not the |←⟩ ⊗ |←⟩ we wanted.

And in fact, the so-called no-cloning theorem implies that it's impossible to duplicate an arbitrary |ψ⟩; the best we can do is copy specific orthogonal states as in the classical case. See also Problem 24B.

§24.4 Deutsch-Jozsa algorithm

The Deutsch-Jozsa algorithm is the first example of a nontrivial quantum algorithm which cannot be performed classically: it is a "proof of concept" that would later inspire Grover's search algorithm and Shor's factoring algorithm.

The problem is as follows: we're given a function f : {0, 1} n → {0, 1}, and promised that the function f is either

• A constant function, or

• A balanced function, meaning that exactly half the inputs map to 0 and half the inputs map to 1.

The function f is given in the form of a reversible black box U f which is the control of a NOT gate, so it can be represented as the circuit diagram

|x 1 x 2 . . . x n ⟩ / n U f |x 1 x 2 . . . x n ⟩ |y⟩ |y + f (x) mod 2⟩ i.e. if f (x 1 , . . . ,

x n ) = 0 then the gate does nothing, otherwise the gate flips the y bit at the bottom. The slash with the n indicates that the top of the input really consists of n qubits, not just the one qubit drawn, and so the black box U f is a map on n + 1 qubits.

The problem is to determine, with as few calls to the black box U f as possible, whether f is balanced or constant.

Question 24.4.1. Classically, show that in the worst case we may need up to 2 n-1 + 1 calls to the function f to answer the question.

So with only classical tools, it would take O(2 n ) queries to determine whether f is balanced or constant. However, Theorem 24.4.2 (Deutsch-Jozsa)

The Deutsch-Jozsa problem can be determined in a quantum circuit with only a single call to the black box.

Proof. For concreteness, we do the case n = 1 explicitly; the general case is contained in Problem 24C. We claim that the necessary circuit is

|0⟩ H U f H |1⟩ H

Here the H's are Hadamard gates, and the meter at the end of the rightmost wire indicates that we make a measurement along the usual |0⟩, |1⟩ basis. This is not a typo! Even though classically the top wire is just a repeat of the input information, we are about to see that it's the top we want to measure.

Note that after the two Hadamard operations, the state we get is

|01⟩ H ⊗2 ---→ 1 √ 2 (|0⟩ + |1⟩) ⊗ 1 √ 2 (|0⟩ -|1⟩) = 1 2 |0⟩ ⊗ |0⟩ -|1⟩ + |1⟩ ⊗ |0⟩ -|1⟩ .

So after applying U f , we obtain

1 2 |0⟩ ⊗ |0 + f (0)⟩ -|1 + f (0)⟩ + |1⟩ ⊗ |0 + f (1)⟩ -|1 + f (1)⟩

where the modulo 2 has been left implicit. Now, observe that the effect of going from

|0⟩ -|1⟩ to |0 + f (x)⟩ -|1 + f (x)⟩ is merely to either keep the state the same (if f (x) = 0) or to negate it (if f (x) = 1)

. So we can simplify and factor to get

1 2 (-1) f (0) |0⟩ + (-1) f (1) |1⟩ ⊗ (|0⟩ -|1⟩) .

Thus, the picture so far is:

|0⟩ H U f 1 √ 2 (-1) f (0) |0⟩ + (-1) f (1) |1⟩ |1⟩ H 1 √ 2 (|0⟩ -|1⟩)

In particular, the resulting state is not entangled, and we can simply discard the last qubit (!). Now observe:

• If f is constant, then the upper-most state is ± |→⟩. • If f is balanced, then the upper-most state is ± |←⟩.

So simply doing a measurement along σ x will give us the answer. Equivalently, perform another H gate (so that H |→⟩ = |0⟩, H |←⟩ = |1⟩) and measuring along σ z in the usual |0⟩, |1⟩ basis. Thus for n = 1 we only need a single call to the oracle. §24.5 A few harder problems to think about Problem 24A (Fredkin gate). The Fredkin gate (also called the controlled swap, or CSWAP gate) is the three-bit gate with the following truth table:

In Out 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1

Thus the gate swaps the last two input bits whenever the first bit is 1. Show that this gate is also reversible and universal.

Problem 24B (Baby no-cloning theorem). Show that there is no unitary map U on two qubits which sends U (|ψ⟩ ⊗ |0⟩) = |ψ⟩ ⊗ |ψ⟩ for any qubit |ψ⟩, i.e. the following circuit diagram is impossible.

|ψ⟩ U

|ψ⟩ |0⟩ |ψ⟩

Problem 24C (Deutsch-Jozsa). Given the black box U f described in the Deutsch-Jozsa algorithm, consider the following circuit.

|0 . . . 0⟩ / n H ⊗n U f H ⊗n |1⟩ H

That is, take n copies of |0⟩, apply the Hadamard rotation to all of them, apply U f , reverse the Hadamard to all n input bits (again discarding the last bit), then measure all n bits in the |0⟩/|1⟩ basis (as in Example 23.4.4). Show that the probability of measuring |0 . . . 0⟩ is 1 if f is constant and 0 if f is balanced.

Problem 24D † (Barenco et al, 1995; arXiv:quant-ph/9503016v1). Let

P = 1 0 0 i Q = 1 √ 2 1 -i -i 1

Verify that the quantum Toffoli gate can be implemented using just controlled rotations via the circuit

|x 1 ⟩ • • • • |x 2 ⟩ • P • |x 3 ⟩ Q Q Q †

This was a big surprise to researchers when discovered, because classical reversible logic requires three-bit gates (e.g. Toffoli, Fredkind).

25 Shor's algorithm

OK, now for Shor's Algorithm: how to factor M = pq in O (log M ) 2 time. This is arguably the reason agencies such as the US's National Security Agency have been diverting millions of dollars toward quantum computing.

§25.1 The classical (inverse) Fourier transform

The "crux move" in Shor's algorithm is the so-called quantum Fourier transform. The Fourier transform is used to extract periodicity in data, and it turns out the quantum analogue is a lot faster than the classical one.

Let me throw the definition at you first. Let N be a positive integer, and let ω N = exp 2πi N .

Definition 25.1.1. Given a tuple of complex numbers

(x 0 , x 1 , . . . , x N -1 )

its discrete inverse Fourier transform is the sequence (y 0 , y 1 , . . . , y N -1 ) defined by

y k = 1 N N -1 j=0 ω jk N x j .

Equivalently, one is applying the matrix

1 N          1 1 1 . . . 1 1 ω N ω 2 N . . . ω N -1 N 1 ω 2 N ω 4 N . . . ω 2(N -1) N . . . . . . . . . . . . . . . 1 ω N -1 N ω 2(N -1) N . . . ω (N -1) 2 N                x 0 x 1 . . . x N -1       =       y 0 y 1 . . . y N -1       .

The reason this operation is important is because it lets us detect if the x i are periodic. More generally, given a sequence of 1's appearing with period r, the amplitudes will peak at inputs which are divisible by N gcd(N,r) . Mathematically, we have that

x k = N -1 j=0 y j ω -jk N .

Example 25. 1 

.2 (Example of discrete inverse Fourier transform)

Let N = 6, ω = ω 6 = exp( 2πi 6 ) and suppose (x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) = (0, 1, 0, 1, 0, 1)

273

(hence x i is periodic modulo 2). Thus,

y 0 = 1 6 ω 0 + ω 0 + ω 0 = 1/2 y 1 = 1 6 ω 1 + ω 3 + ω 5 = 0 y 2 = 1 6 ω 2 + ω 6 + ω 10 = 0 y 3 = 1 6 ω 3 + ω 9 + ω 15 = -1/2 y 4 = 1 6 ω 4 + ω 12 + ω 20 = 0 y 5 = 1 6 ω 5 + ω 15 + ω 25 = 0.

Thus, in the inverse transformation the "amplitudes" are all concentrated at multiples of 3; thus this reveals the periodicity of the original sequence by N 3 = 2.

Remark 25.1.3 -The fact that this operation is called the "inverse" Fourier transform is mostly a historical accident (as my understanding goes). Confusingly, the corresponding quantum operation is the (not-inverted) Fourier transform.

If we apply the definition as written, computing the transform takes O(N 2 ) time. It turns out that by a classical algorithm called the fast Fourier transform (whose details we won't discuss, but it effectively "reuses" calculations), one can reduce this to O(N log N ) time. However, for Shor's algorithm this is also insufficient; we need something like O (log N ) 2 instead. This is where the quantum Fourier transform comes in.

§25.2 The quantum Fourier transform

Note that to compute a Fourier transform, we need to multiply an N × N matrix with an N -vector, so this takes O(N 2 ) multiplications. However, we are about to show that with a quantum computer, one can do this using O((log N ) 2 ) quantum gates when N = 2 n , on a system with n qubits. First, some more notation:

Abuse of Notation 25.2. 1. In what follows, |x⟩ will refer to

|x n ⟩ ⊗ |x n-1 ⟩ ⊗ • • • ⊗ |x 1 ⟩ where x = x n x n-1 . . . x 1 in binary. For example, if n = 3 then |6⟩ really means |1⟩⊗|1⟩⊗|0⟩.

Likewise, we refer to 0.x 1 x 2 . . . x n as binary.

Observe that the n-qubit space now has an orthonormal basis |0⟩, |1⟩, . . 

= N -1 k=0 x k |k⟩ .

The quantum Fourier transform is defined by

U QFT (|ψ⟩) = 1 √ N N -1 j=0 N -1 k=0 ω jk N x k |j⟩ .

In other words, using the basis |0⟩, . . . , |N -1⟩, U QFT is given by the matrix

U QFT = 1 √ N          1 1 1 . . . 1 1 ω N ω 2 N . . . ω N -1 N 1 ω 2 N ω 4 N . . . ω 2(N -1) N . . . . . . . . . . . . . . . 1 ω N -1 N ω 2(N -1) N . . . ω (N -1) 2 N         

This is the exactly the same definition as before, except we have a √ N factor added so that U QFT is unitary. But the trick is that in the quantum setup, the matrix can be rewritten:

Proposition 25.2.3 (Tensor representation) Let |x⟩ = |x n x n-1 . . . x 1 ⟩. Then U QFT (|x n x n-1 . . . x 1 ⟩) = 1 √ N (|0⟩ + exp(2πi • 0.x 1 ) |1⟩) ⊗ (|0⟩ + exp(2πi • 0.x 2 x 1 ) |1⟩) ⊗ . . . ⊗ (|0⟩ + exp(2πi • 0.x n . . . x 1 ) |1⟩)

Proof. Direct (and quite annoying) computation. In short, expand everything.

So by using mixed states, the quantum Fourier transform can use this "multiplication by tensor product" trick that isn't possible classically. Now, without further ado, here's the circuit. Define the rotation matrices

R k = 1 0 0 exp(2πi/2 k ) .

Then, for n = 3 the circuit is given by using controlled R k 's as follows:

|x 3 ⟩ H R 2 R 3 |y 1 ⟩ |x 2 ⟩ • H R 2 |y 2 ⟩ |x 1 ⟩ • • H |y 3 ⟩ Exercise 25.2.4. Show that in this circuit, the image of |x 3 x 2 x 1 ⟩ is |0⟩ + exp(2πi • 0.x 1 ) |1⟩ ⊗ |0⟩ + exp(2πi • 0.x 2 x 1 ) |1⟩ ⊗ |0⟩ + exp(2πi • 0.x 3 x 2 x 1 ) |1⟩ as claimed.

For general n, we can write this as inductively as

|x n ⟩ QFT n-1 R n • • • • • • |y 1 ⟩ |x n-1 ⟩ R n-1 • • • • • • |y 2 ⟩ . . . . . . |x i ⟩ • • • R i • • • |y n-i+1 ⟩ . . . . . . |x 2 ⟩ • • • • • • R 2 |y n-1 ⟩ |x 1 ⟩ • • • • • • • • • • H |y n ⟩

Question 25.2.5. Convince yourself that when n = 3 the two circuits displayed are equivalent.

Thus, the quantum Fourier transform is achievable with O(n 2 ) gates, which is enormously better than the O(N log N ) operations achieved by the classical fast Fourier transform (where N = 2 n ).

§25.3 Shor's algorithm

The quantum Fourier transform is the key piece of Shor's algorithm. Now that we have it, we can solve the factoring problem.

Let p, q > 3 be odd primes, and assume p ̸ = q. The main idea is to turn factoring an integer M = pq into a problem about finding the order of x (mod M ); the latter is a "periodicity" problem that the quantum Fourier transform will let us solve. Specifically, say that an x (mod M ) is good if

(i) gcd(x, M ) = 1,

(ii) The order r of x (mod M ) is even, and

(iii) Factoring 0 ≡ (x r/2 -1)(x r/2 +1) (mod M ), neither of the two factors is 0 (mod M ).

Thus one of them is divisible by p, and the other is divisible by q.

Exercise 25.3.1 (For contest number theory practice). Show that for M = pq at least half of the residues in (Z/M Z) × are good.

So if we can find the order of an arbitrary x ∈ (Z/M Z) × , then we just keep picking x until we pick a good one (this happens more than half the time); once we do, we compute gcd(x r/2 -1, M ) using the Euclidean algorithm to extract one of the prime factors of M , and we're home free.

Now how do we do this? The idea is not so difficult: first we generate a sequence which is periodic modulo r.

Example 25.3.2 (Factoring 77: generating the periodic state) Let's say we're trying to factor M = 77, and we randomly select x = 2, and want to find its order r. Let n = 13 and N = 2 13 , and start by initializing the state

|ψ⟩ = 1 √ N N -1 k=0 |k⟩ . Now, build a circuit U x (depending on x) which takes |k⟩ |0⟩ to |k⟩ |x k mod M ⟩. Applying this to |ψ⟩ ⊗ |0⟩ gives U (|ψ⟩ |0⟩) = 1 √ N N -1 k=0 |k⟩ ⊗ |x k mod M ⟩ .

Now suppose we measure the second qubit, and get a state of |128⟩. That tells us that the collapsed state now, up to scaling, is

(|7⟩ + |7 + r⟩ + |7 + 2r⟩ + . . . ) ⊗ |128⟩ .

The bottleneck is actually the circuit U x ; one can compute x k (mod M ) by using repeated squaring, but it's still the clumsy part of the whole operation.

In general, the operation is:

• Pick a sufficiently large N = 2 n (say, N ≥ M 2 ). • Generate |ψ⟩ = 2 n -1 k=0 |k⟩. • Build a circuit U x which computes |x k mod M ⟩. • Apply it to get a state 1 √ N 2 n -1 k=0 |k⟩ ⊗ |x k mod M ⟩.

• Measure the second qubit to cause the first qubit to collapse to something which is periodic modulo r. Let |ϕ⟩ denote the left qubit.

Suppose we apply the quantum Fourier transform to the left qubit |ϕ⟩ now: since the left bit is periodic modulo r, we expect the transform will tell us what r is. Unfortunately, this doesn't quite work out, since N is a power of two, but we don't expect r to be.

Nevertheless, consider a state

|ϕ⟩ = |k 0 ⟩ + |k 0 + r⟩ + . . .

so for example previously we had k 0 = 7 if we measured 128 on x = 2. Applying the quantum Fourier transform, we see that the coefficient of |j⟩ in the transformed image is equal to This won't work all the time1 (for example, we could get unlucky and measure j = 0, i.e. s = 0, which would tell us no information at all).

ω k 0 j N • ω 0 N + ω jr N + ω 2jr N + ω 3jr N + . . .

But one can show that we succeed any time that gcd(s, r) = 1.

This happens at least 1 log r of the time, and since r < M this means that given sufficiently many trials, we will eventually extract the correct order r. This is Shor's algorithm.

VIII Calculus 101

Part VIII: Contents 

27

Bonus: A hint of p-adic numbers 293 27.1 

28

Differentiation 305 28.1 

29

Power series and Taylor series 317 29.1 29.6 This all works over complex numbers as well, except also complex analysis is heaven . . 29.7 A few harder problems to think about .

30

Riemann integrals 325 30.1 

. Now that we have developed the theory of metric (and topological) spaces well, we give a three-chapter sequence which briskly covers the theory of single-variable calculus.

Much of the work has secretly already been done, For example, if x n and y n are real sequences with lim n x n = x and lim n y n = y, then in fact lim n (x n + y n ) = x + y or lim n (x n y n ) = xy, because we showed in Proposition 2.5.5 that arithmetic was continuous. We will also see that completeness plays a crucial role.

§26.1 Completeness and inf/sup

Prototypical example for this section:

sup[0, 1] = sup(0, 1) = 1.

As R is a metric space, we may discuss continuity and convergence. There are two important facts about R which will make most of the following sections tick.

The first fact you have already seen before:

Theorem 26.1.1 (R is complete)

As a metric space, R is complete: sequences converge if and only if they are Cauchy.

The second one we have not seen before -it is the existence of inf and sup. Your intuition should be: sup is max adjusted slightly for infinite sets. (And inf is adjusted min.)

Why the "adjustment"? Example 26.1.2 (Why is max not good enough?) Let's say we have the open interval S = (0, 1). The elements can get arbitrarily close to 1, so we would like to think "1 is the max of S"; except the issue is that 1 / ∈ S. In general, infinite sets don't necessarily have a maximum, and we have to talk about bounds instead.

So we will define sup S in such a way that sup S = 1. The definition is that "1 is the smallest number which is at least every element of S". The sequence a n = 1 n has infimum inf a n = 0. §26.2 Proofs of the two key completeness properties of R Careful readers will note that we have not actually proven either Theorem 26. 1.4 or Theorem 26.1.1. We will do so here. First, we show that the ability to take infimums and supremums lets you prove completeness of R.

Proof that Theorem 26. 1.4 implies Theorem 26. 1.1. Let a 1 , a 2 , . . . be a Cauchy sequence. By discarding finitely many leading terms, we may as well assume that |a i -a j | ≤ 100 for all i and j. In particular, the sequence is now bounded; it lies between [a 1 -100, a 1 + 100] for example.

We want to show this sequence converges, so we have to first describe what the limit is. We know that to do this we are really going to have to use the fact that we live in R.

(For example we know in Q the limit of 1, 1.4, 1.41, 1.414, . . . is nonexistent.)

We propose the following: let

S = {x ∈ R | a n ≥ x for infinitely many n } .

We claim that the sequence converges to M = sup S.

Exercise 26.2.1. Show that this supremum makes sense by proving that a 1 -100 ∈ S (so S is nonempty) while all elements of S are at most a 1 + 100 (so S is bounded above). Thus we are allowed to actually take the supremum.

You can think of this set S with the following picture. We have a Cauchy sequence drawn in the real line which we think converges, which we can visualize as a bunch of dots on the real line, with some order on them. We wish to cut the line with a knife such that only finitely many dots are do to the left of the knife. (For example, placing the knife all the way to the left always works.) The set S represents the places where we could put the knife, and M is "as far right" as we could go. Because of the way supremums work, M might not itself be a valid knife location, but certainly anything to its left is.

R M M -1 2 ε a 1 a 2 a 3 a 4 a 5 a 6 a 8 a 7

Let ε > 0 be given; we want to show eventually all terms are within ε of M . Because the sequence is Cauchy, there is an N such that eventually |a m -a n | < 1 2 ε for m ≥ n ≥ N . Now suppose we fix n and vary m. By the definition of M , it should be possible to pick the index m such that a m ≥ M -1 2 ε (there are infinitely many to choose from since M -1 2 ε is a valid knife location, and we only need m ≥ n). In that case we have

|a n -M | ≤ |a n -a m | + |a m -M | < 1 2 ε + 1 2 ε = ε

by the triangle inequality. This completes the proof.

Therefore it is enough to prove the latter Theorem 26.1.4. To do this though, we would need to actually give a rigorous definition of the real numbers R, since we have not done so yet! One approach that makes this easy is to use the so-called Dedekind cut construction. Suppose we take the rational numbers Q. Then one defines a real number to be a "cut" A | B of the set of rational numbers: a pair of subsets of Q such that

• Q = A ⊔ B is a disjoint union;

• A and B are nonempty;

• we have a < b for every a ∈ A and b ∈ B, and

• A has no largest element (i.e. sup A / ∈ A).

This can again be visualized by taking what you think of as the real line, and slicing at some real number. The subset Q ⊂ R gets cut into two halves A and B. If the knife happens to land exactly at a rational number, by convention we consider that number to be in the right half (which explains the last fourth condition that sup A / ∈ A). With this definition Theorem 26. 1.4 is easy: to take the supremum of a set of real numbers, we take the union of all the left halves. The hard part is then figuring out how to define +, -, ×, ÷ and so on with this rather awkward construction. If you want to read more about this construction in detail, my favorite reference is [Pu02] , in which all of this is done carefully in Chapter 1.

§26.3 Monotonic sequences

Here is a great exercise. The proof here readily adapts by shifting. and so on, where in general we stuck on the decimal representation of the next power of 2. This will converge to some real number, although of course this number is quite unnatural and there is probably no good description for it.

Definition 26.3.2. A sequence a n is monotonic if either a 1 ≥ a 2 ≥ . . . or a 1 ≤ a 2 ≤ . . . .

In general, "infinite decimals" can now be defined as the limit of the truncated finite ones.

Example 26.3.5 (0.9999

• • • = 1)

In particular, I can finally make precise the notion you argued about in elementary school that 0.9999

• • • = 1.

We simply define a repeating decimal to be the limit of the sequence 0.9, 0.99, 0.999 . . . . And it is obvious that the limit of this sequence is 1.

Some of you might be a little surprised since it seems like we really should have 0.9999 = 9 • 10 -1 + 9 • 10 -2 + . . . -the limit of "partial sums". Don't worry, we're about to define those in just a moment.

Here is one other great use of monotonic sequences. 

a n := lim N →∞ sup n≥N a n = lim N →∞ sup {a N , a N +1 , . . . } .

This is called the limit supremum of (a n ). We set lim sup n→∞ a n to be +∞ if a n is not bounded above.

If a n is bounded above, the limit infimum lim inf n→∞ a n is defined similarly. In particular, lim inf n→∞ a n = -∞ if a n is not bounded below.

Exercise 26.3.7. Show that these definitions make sense, by checking that the supremums are non-increasing, and bounded below.

We can think of lim sup n a n as "supremum, but allowing finitely many terms to be discarded". §26.4 Infinite series Prototypical example for this section:

∞ k≥1 1 k(k+1) = lim n→∞ 1 -1 n+1 = 1.

We will actually begin by working with infinite series, since in the previous chapters we defined limits of sequences, and so this is actually the next closest thing to work with. 1 This will give you a rigorous way to think about statements like

∞ n=1 1 n 2 = π 2 6

and help answer questions like "how can you add rational numbers and get an irrational one?".

Definition 26.4.1. Consider a sequence a 1 , . . . of real numbers. The series k a k converges to a limit L if the sequence of "partial sums"

s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 . .

.

s n = a 1 + • • • + a n converges to the limit L. Otherwise it diverges.

Abuse of Notation 26.4.2 (Writing divergence as +∞). It is customary, if all the a k are nonnegative, to write k a k = ∞ to denote that the series diverges.

You will notice that by using the definition of sequences, we have masterfully sidestepped the issue of "adding infinitely many numbers" which would otherwise cause all sorts of problems.

An "infinite sum" is actually the limit of its partial sums. There is no infinite addition involved.

That's why it's for example okay to have n≥1

1 n 2 = π 2

6 be irrational; we have already seen many times that sequences of rational numbers can converge to irrational numbers. It also means we can gladly ignore all the irritating posts by middle schoolers about

1 + 2 + 3 + • • • = -1 12

; the partial sums explode to +∞, end of story, and if you want to assign a value to that sum it had better be a definition.

Example 26.4.3 (The classical telescoping series)

We can now prove the classic telescoping series

∞ k=1 1 k(k + 1)

in a way that doesn't just hand-wave the ending. Note that the kth partial sum is

n k=1 1 k(k + 1) = 1 1 • 2 + 1 2 • 3 + • • • + 1 n(n + 1) = 1 1 - 1 2 + • • • + 1 n - 1 n + 1 = 1 - 1 n + 1 .

The limit of this partial sum as n → ∞ is 1.

Example 26.4.4 (Harmonic series diverges)

We can also make sense of the statement that ∞ k=1 1 k = ∞ (i.e. it diverges). We may bound the 2 n th partial sums from below:

2 n k=1 1 k = 1 1 + 1 2 + • • • + 1 2 n ≥ 1 1 + 1 2 + 1 4 + 1 4 + 1 8 + 1 8 + 1 8 + 1 8 + • • • + 1 2 n + • • • + 1 2 n 2 n-1 terms = 1 + 1 2 + 1 2 + • • • + 1 2 = 1 + n -1 2 .

A sequence satisfying s 2 n ≥ 1 + 1 2 (n -1) will never converge to a finite number! I had better also mention that for nonnegative sums, convergence is just the same as having "finite sum" in the following sense.

Proposition 26.4.5 (Partial sums of nonnegatives bounded implies convergent)

Let k a k be a series of nonnegative real numbers. Then k a k converges to some limit if and only if there is a constant M such that

a 1 + • • • + a n < M

for every positive integer n.

Proof. This is actually just Theorem 26.3.3 in disguise, but since we left the proof as an exercise back then, we'll write it out this time.

Obviously if no such M exists then convergence will not happen, since this means the sequence s n of partial sums is unbounded.

Conversely, if such M exists then we have

s 1 ≤ s 2 ≤ • • • < M .

Then we contend the sequence s n converges to L := sup n s n < ∞. (If you read the proof that completeness implies Cauchy, the picture is nearly the same here, but simpler.)

R L L -ε s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8

Indeed, this means for any ε there are infinitely many terms of the sequence exceeding L -ε; but since the sequence is monotonic, once 

s n ≥ L -ε then s n ′ ≥ L -ε for all n ′ ≥ n.

k a k < ∞

as a shorthand for " k a k converges to a finite limit", (or perhaps shorthand for " k a k is bounded" -as we have just proved these are equivalent). We will use this notation too. §26.5 Series addition is not commutative: a horror story One unfortunate property of the above definition is that it actually depends on the order of the elements. In fact, it turns out that there is an explicit way to describe when rearrangement is okay. Then, rearrangement works great.

Theorem 26.5.4 (Permutation of terms okay for absolute convergence)

Consider a series k a k which is absolutely convergent and has limit L. Then any permutation of the terms will also converge to L.

Proof. Suppose k a k converges to L, and b n is a rearrangement. Let ε > 0. We will show that the partial sums of b n are eventually within ε of L.

The hypothesis means that there is a large N in terms of ε such that

N k=1 a k -L < 1 2 ε and n k=N +1 |a k | < 1 2 ε

for every n ≥ N (the former from vanilla convergence of a k and the latter from the fact that a k converges absolutely, hence its partial sums are Cauchy). Now suppose M is large enough that a 1 , . . . , a N are contained within the terms

{b 1 , . . . , b M }. Then b 1 + • • • + b M = (a 1 + • • • + a N ) + a i 1 + a i 2 + • • • + a i M -N M -N terms with indices ≥ N

The terms in the first line sum up to within 1 2 ε of L, and the terms in the second line have sum at most 1 2 ε in absolute value, so the total b

1 + • • • + b M is within 1 2 ε + 1 2 ε = ε of L.

In particular, when you have nonnegative terms, the world is great: Nonnegative series can be rearranged at will.

And the good news is that actually, in practice, most of your sums will be nonnegative.

The converse is not true, and in fact, it is almost the worst possible converse you can imagine.

Theorem 26.5.5 (Riemann rearrangement theorem: Permutation of terms meaningless for conditional convergence) Consider a series k a k which converges conditionally to some real number. Then, there exists a permutation of the series which converges conditionally to 1337.

(Or any constant. You can also get it to diverge, too.)

So, permutation is as bad as possible for conditionally convergent series, and hence don't even bother to try.

§26.6 Limits of functions at points

Prototypical example for this section: lim x→∞ 1/x = 0.

We had also better define the notion of a limit of a real function, which (surprisingly) we haven't actually defined yet. The definition will look like what we have seen before with continuity. Definition 26.6.1. Let f : R → R be a function2 and let p ∈ R be a point in the domain. Suppose there exists a real number L such that:

For every ε > 0, there exists δ > 0 such that if |x -p| < δ and x ̸ = p then |f (x) -L| < ε.

Then we say L is the limit of f as x → p, and write

lim x→p f (x) = L.

There is an important point here: in this definition we deliberately require that x ̸ = p.

The value lim x→p f (x) does not depend on f (p), and accordingly we often do not even bother to define f (p). 

f (x) = 3x if x ̸ = 0 2019 otherwise.

Then lim x→0 f (x) = 0. The value f (0) = 2019 does not affect the limit. Obviously, because f (0) was made up to be some artificial value that did not agree with the limit, this function is discontinuous at x = 0. 

f (x) =        1 if x = 0 1 q if x = p

q where q > 0 and gcd(p,

q) = 1 0 if x / ∈ Q. For example, f (π) = 0, f (2/3) = 1 3 , f (0.17) = 1 100 . Then lim x→0 f (x) = 0.

For example, if |x| < 1/100 and x ̸ = 0 then f (x) is either zero (for x irrational) or else is at most 1 101 (if x is rational). As f (0) = 1, this function is also discontinuous at x = 0. However, if we change the definition so that f (0) = 0 instead, then f becomes continuous at 0. We will not prove this here, since I don't want to get into trig yet. In general, I will basically only use trig functions for examples and not for any theory, so most properties of the trig functions will just be quoted.

Abuse of Notation 26.6.6 (The usual notation). From now on, the above example will usually be abbreviated to just

lim x→0 sin x x = 1.

The reason there is a slight abuse here is that I'm supposed to feed a function f into the limit, and instead I've written down an expression which is defined everywhere -except at x = 0. But that f (0) value doesn't change anything. So the above means: "the limit of the function described by f (x) = sin x x , except f (0) can be whatever it wants because it doesn't matter". Remark 26.6.7 (For metric spaces) -You might be surprised that I didn't define the notion of lim x→p f (x) earlier for f : M → N a function on metric spaces. We can actually do so as above, but there is one nuance: what if our metric space M is discrete, so p has no points nearby it? (Or even more simply, what if M is a one-point space?) We then cannot define lim x→p f (x) at all.

Thus if f : M → N and we want to define lim x→p f (x), we have the requirement that p should have a point within ε of it, for any ε > 0. In other words, p should not be an isolated point.

As usual, there are no surprises with arithmetic, we have lim x→p (f (x) ± g(x)) = lim x→p f (x) ± lim x→p g(x), and so on and so forth. We have effectively done this proof before so we won't repeat it again. §26.7 Limits of functions at infinity Annoyingly, we actually have to make this definition separately, even though it will not feel any different from earlier examples. For every ε > 0, there exists a constant M such that if x > M , then |f (x) -L| < ε.

Then we say L is the limit of f as x approaches ∞ and write

lim x→∞ f (x) = L. The limit lim x→-∞ f (x) is defined similarly, with x > M replaced by x < M .

Fortunately, as ∞ is not an element of R, we don't have to do the same antics about f (∞) like we had to do with "f (p) set arbitrarily". So these examples can be more easily written down. I'll even write out the proof: for any ε > 0, if x > 1/ε then 1 x -0 < ε.

There are no surprises with arithmetic: we have lim x→∞ (f (x)±g(x)) = lim x→∞ f (x)± lim x→p g(x), and so on and so forth. This is about the fourth time I've mentioned this, so I will not say more. §26.8 A few harder problems to think about Problem 26A. Define the sequence

a n = (-1) n + n 3 2 n

for every positive integer n. Compute the limit infimum and the limit supremum. 

f (x) =        1 if x = 0 1 q if x = p q where q > 0 and gcd(p, q) = 1 0 if x / ∈ Q.

For every real number p, compute lim x→p f (x), if it exists. At which points is f continuous?

27 Bonus: A hint of p-adic numbers This is a bonus chapter meant for those who have also read about rings and fields: it's a nice tidbit at the intersection of algebra and analysis.

In this chapter, we are going to redo most of the previous chapter with the absolute value |-| replaced by the p-adic one. This will give us the p-adic integers Z p , and the p-adic numbers Q p . The one-sentence description is that these are "integers/rationals carrying full mod p e information" (and only that information).

In everything that follows p is always assumed to denote a prime. The first four sections will cover the founding definitions culminating in a short solution to a USA TST problem. We will then state (mostly without proof) some more surprising results about continuous functions f : Z p → Q p ; finally we close with the famous proof of the Skolem-Mahler-Lech theorem using p-adic analysis. §27. In elementary/olympiad number theory, we're already well-familiar with the following two ideas:

• Taking modulo a prime p or prime power p e , and • Looking at the exponent ν p .

Let me expand on the first point. Suppose we have some Diophantine equation. In olympiad contexts, one can take an equation modulo p to gain something else to work with. Unfortunately, taking modulo p loses some information: the reduction Z ↠ Z/p is far from injective.

If we want finer control, we could consider instead taking modulo p 2 , rather than taking modulo p. This can also give some new information (cubes modulo 9, anyone?), but it has the disadvantage that Z/p 2 isn't a field, so we lose a lot of the nice algebraic properties that we got if we take modulo p.

One of the goals of p-adic numbers is that we can get around these two issues I described. The p-adic numbers we introduce is going to have the following properties:

1. You can "take modulo p e for all e at once". In 

f p (x) = p-1 k=1 1 (px + k) 2 (mod p 3 ) does not depend on x.

Here is a problem where we clearly only care about p e -type information. Yet it's a nontrivial challenge to do the necessary manipulations mod p 3 (try it!). The basic issue is that there is no good way to deal with the denominators modulo p 3 (in part Z/p 3 is not even an integral domain).

However, with p-adic analysis we're going to be able to overcome these limitations and give a "straightforward" proof by using the identity

1 + px k -2 = n≥0 -2 n px k n .

Such an identity makes no sense over Q or R for convergence reasons, but it will work fine over Q p , which is all we need.

§27.2 Algebraic perspective

Prototypical example for this section:

-1/2 = 1 + 3 + 3 2 + 3 3 + • • • ∈ Z 3 .

We now construct Z p and Q p . I promised earlier that a p-adic integer will let you look at "all residues modulo p e " at once. This definition will formalize this. §27.2.i Definition of Z p Definition 27.2.1 (Introducing Z p ). A p-adic integer is a sequence x = (x 1 mod p, x 2 mod p2 , x 3 mod p 3 , . . . ) of residues x e modulo p e for each integer e, satisfying the compatibility relations x i ≡ x j (mod p i ) for i < j.

The set Z p of p-adic integers forms a ring under component-wise addition and multiplication.

Example 27.2.2 (Some 3-adic integers)

Let p = 3. Every usual integer n generates a (compatible) sequence of residues modulo p e for each e, so we can view each ordinary integer as p-adic one: 50 = (2 mod 3, 5 mod 9, 23 mod 27, 50 mod 81, 50 mod 243, . . . ) .

On the other hand, there are sequences of residues which do not correspond to any usual integer despite satisfying compatibility relations, such as (1 mod 3, 4 mod 9, 13 mod 27, 40 mod 81, . . . ) which can be thought of as x = 1 + p + p 2 + . . . .

In this way we get an injective map

Z → Z p n → n mod p, n mod p 2 , n mod p 3 , . . .

which is not surjective. So there are more p-adic integers than usual integers.

(Remark for experts: those of you familiar with category theory might recognize that this definition can be written concisely as

Z p := lim ← - Z/p e Z

where the inverse limit is taken across e ≥ 1.)

Exercise 27.2.3. Check that Z p is an integral domain.

§27.2.ii Base p expansion

Here is another way to think about p-adic integers using "base p". As in the example earlier, every usual integer can be written in base p, for example

50 = 1212 3 = 2 • 3 0 + 1 • 3 1 + 2 • 3 2 + 1 • 3 3 .

More generally, given any x = (x 1 , . . . ) ∈ Z p , we can write down a "base p" expansion in the sense that there are exactly p choices of x k given x k-1 . Continuing the example earlier, we would write 

. ).)

Thus you may often hear the advertisement that a p-adic integer is a "possibly infinite base p expansion". This is correct, but later on we'll be thinking of Z p in a more and more "analytic" way, and so I prefer to think of this as p-adic integers are Taylor series with base p.

Indeed, much of your intuition from generating functions K[[X]]

(where K is a field) will carry over to Z p . §27.2.iii Constructing Q p Here is one way in which your intuition from generating functions carries over: Proposition 27.2.4 (Non-multiples of p are all invertible)

The number x ∈ Z p is invertible if and only if x 1 ̸ = 0. In symbols,

x ∈ Z × p ⇐⇒ x ̸ ≡ 0 (mod p).

Contrast this with the corresponding statement for

K[[X]]: a generating function F ∈ K[[X]] is invertible iff F (0) ̸ = 0.

Proof. If x ≡ 0 (mod p) then x 1 = 0, so clearly not invertible. Otherwise, x e ̸ ≡ 0 (mod p) for all e, so we can take an inverse y e modulo p e , with x e y e ≡ 1 (mod p e ). As the y e are themselves compatible, the element (y 1 , y 2 , . . . ) is an inverse. (Compare this with the "geometric series" 1 With this observation, here is now the definition of Q p . Definition 27.2.8 (Introducing Q p ). Since Z p is an integral domain, we let Q p denote its field of fractions. These are the p-adic numbers.

+ 3 + 3 2 + • • • = 1 1-3

Continuing our generating functions analogy:

Z p is to Q p as K[[X]] is to K((X)).

This means

Q p can be thought of as Laurent series with base p. and in particular according to the earlier proposition we deduce: Proposition 27.2.9 (Q p looks like formal Laurent series)

Every nonzero element of Q p is uniquely of the form p k u where k ∈ Z, u ∈ Z × p .

Thus, continuing our base p analogy, elements of Q p are in bijection with "Laurent series" k≥-n To keep pushing the formal Laurent series analogy, K((X)) is usually not thought of as quotient of generating functions but rather as "formal series with some negative exponents". You should apply the same intuition on Q p . Remark 27.2.12 -At this point I want to make a remark about the fact 1/p ∈ Q p , connecting it to the wish-list of properties I had before. In elementary number theory you can take equations modulo p, but if you do the quantity n/p mod p doesn't make sense unless you know n mod p 2 . You can't fix this by just taking modulo p 2 since then you need n mod p 3 to get n/p mod p 2 , ad infinitum. You can work around issues like this, but the nice feature of Z p and Q p is that you have modulo p e information for "all e at once": the information of x ∈ Q p packages all the modulo p e information simultaneously. So you can divide by p with no repercussions. §27.3 Analytic perspective §27. 3 

a k p k = . . . a 2 a 1 a 0 .a -1 a -2 .

.i Definition

Up until now we've been thinking about things mostly algebraically, but moving forward it will be helpful to start using the language of analysis. Usually, two real numbers are considered "close" if they are close on the number of line, but for p-adic purposes we only care about modulo p e information. So, we'll instead think of two elements of Z p or Q p as "close" if they differ by a large multiple of p e .

For this we'll borrow the familiar ν p from elementary number theory.

Definition 27.3.1 (p-adic valuation and absolute value). We define the p-adic valuation ν p : Q × p → Z in the following two equivalent ways:

• For x = (x 1 , x 2 , . . . ) ∈ Z p we let ν p (x) be the largest e such that x e ≡ 0 (mod p e ) (or e = 0 if x ∈ Z × p ). Then extend to all of Q × p by ν p (xy) = ν p (x) + ν p (y).

• Each x ∈ Q × p can be written uniquely as

p k u for u ∈ Z × p , k ∈ Z. We let ν p (x) = k.

By convention we set ν p (0) = +∞. Finally, define the p-adic absolute value |•| p by

|x| p = p -νp(x) .

In particular |0| p = 0.

This fulfills the promise that x and y are close if they look the same modulo p e for large e; in that case ν p (x -y) is large and accordingly |x -y| p is small. §27.

3.ii Ultrametric space

In this way, Q p and Z p becomes a metric space with metric given by |x -y| p .

Exercise 27.3.2. Suppose f : Z p → Q p is continuous and f (n) = (-1) n for every n ∈ Z ≥0 . Prove that p = 2.

In fact, these spaces satisfy a stronger form of the triangle inequality than you are used to from R. 

Equality holds if (but not only if) |x|

p ̸ = |y| p .

However, Q p is more than just a metric space: it is a field, with its own addition and multiplication. This means we can do analysis just like in R or C: basically, any notion such as "continuous function", "convergent series", et cetera has a p-adic analog. In particular, we can define what it means for an infinite sum to converge: Definition 27.3.4 (Convergence notions). Here are some examples of p-adic analogs of "real-world" notions.

• A sequence s 1 , . . . converges to a limit L if lim n→∞ |s n -L| p = 0.

• The infinite series k x k converges if the sequence of partial sums s 1 = x 1 , s 2 = x 1 + x 2 , . . . , converges to some limit. With this definition in place, the "base p" discussion we had earlier is now true in the analytic sense: if

x = . . . a 2 a 1 a 0p ∈ Z p then ∞ k=0 a k p k converges to x.

Indeed, the difference between x and the nth partial sum is divisible by p n , hence the partial sums approach x as n → ∞.

While the definitions are all the same, there are some changes in properties that should be true. For example, in Q p convergence of partial sums is simpler:

Proposition 27.3.5 (|x k | p → 0 iff convergence of series) A series ∞ k=1 x k in Q p converges to some limit if and only if lim k→∞ |x k | p = 0.

Contrast this with

1 n = ∞ in R.

You can think of this as a consequence of strong triangle inequality.

Proof. By multiplying by a large enough power of p, we may assume x k ∈ Z p . (This isn't actually necessary, but makes the notation nicer.)

Observe that x k (mod p) must eventually stabilize, since for large enough n we have

|x n | p < 1 ⇐⇒ ν p (x n ) ≥ 1.

So let a 1 be the eventual residue modulo p of N k=0 x k (mod p) for large N . In the same way let a 2 be the eventual residue modulo p 2 , and so on. Then one can check we approach the limit a = (a 1 , a 2 , . . . ).

§27.3.iii More fun with geometric series

Let's finally state the p-adic analog of the geometric series formula.

Proposition 27.3.6 (Geometric series)

Let x ∈ Z p with |x| p < 1. Then

1 1 -x = 1 + x + x 2 + x 3 + . . . .

Proof.

Note that the partial sums satisfy 1

+ x + x 2 + • • • + x n = 1-x n 1-x , and x n → 0 as n → ∞ since |x| p < 1. So, 1 + 3 + 3 2 + • • • = -1

2 is really a correct convergence in Z 3 . And so on. If you buy the analogy that Z p is generating functions with base p, then all the olympiad generating functions you might be used to have p-adic analogs. For example, you can prove more generally that: (I haven't defined (1 + x) r , but it has the properties you expect.) §27.

3.iv Completeness

Note that the definition of |•| p could have been given for Q as well; we didn't need Q p to introduce it (after all, we have ν p in olympiads already). The big important theorem I must state now is:

Theorem 27.3.8 (Q p is complete) The space Q p is the completion of Q with respect to |•| p .

This is the definition of Q p you'll see more frequently; one then defines Z p in terms of Q p (rather than vice-versa) according to

Z p = x ∈ Q p : |x| p ≤ 1 . §27.3

.v Philosophical notes

Let me justify why this definition is philosophically nice. Suppose you are an ancient Greek mathematician who is given:

Problem for Ancient Greeks. Estimate the value of the sum

S = 1 1 2 + 1 2 2 + • • • + 1 10000 2 to within 0.001.

The sum S consists entirely of rational numbers, so the problem statement would be fair game for ancient Greece. But it turns out that in order to get a good estimate, it really helps if you know about the real numbers: because then you can construct the infinite series n≥1 n -2 = 1 6 π 2 , and deduce that S ≈ π 2 6 , up to some small error term from the terms past 1 10001 2 , which can be bounded. Of course, in order to have access to enough theory to prove that S = π 2 /6, you need to have the real numbers; it's impossible to do calculus in Q (the sequence 1, 1.4, 1.41 

f p (x) = p-1 k=1 1 (px + k) 2 to within mod p 3 .

Even though f p (x) is a rational number, it still helps to be able to do analysis with infinite sums, and then bound the error term (i.e. take mod p 3 ). But the space Q is not complete with respect to |•| p either, and thus it makes sense to work in the completion of Q with respect to |•| p . This is exactly Q p .

In any case, let's finally solve Example 27.1.1.

Example 27.3.9 (USA TST 2002)

We will now compute

f p (x) = p-1 k=1 1 (px + k) 2 (mod p 3 ).

Armed with the generalized binomial theorem, this becomes straightforward.

f p (x) = p-1 k=1 1 (px + k) 2 = p-1 k=1 1 k 2 1 + px k -2 = p-1 k=1 1 k 2 n≥0 -2 n px k n = n≥0 -2 n p-1 k=1 1 k 2 x k n p n ≡ p-1 k=1 1 k 2 -2x   p-1 k=1 1 k 3   p + 3x 2   p-1 k=1 1 k 4   p 2 (mod p 3 ).

Using the elementary facts that p 2 | k k -3 and p | k k -4 , this solves the problem.

§27.4 Mahler coefficients

One of the big surprises of p-adic analysis is that:

We can basically describe all continuous functions

Z p → Q p .

They are given by a basis of functions

x n := x(x -1) . . . (x -(n -1)) n!

in the following way. Let f : Z p → Q p be continuous, and define

a n = n k=0 n k (-1) n-k f (k). (Then lim n a n = 0 and

f (x) = n≥0 a n x n .

Conversely, if a n is any sequence converging to zero, then f (x) = n≥0 a n

x n defines a continuous function satisfying (27.1) .

The a i are called the Mahler coefficients of f .

Exercise 27.4.2. Last post we proved that if f : Z p → Q p is continuous and f (n) = (-1) n for every n ∈ Z ≥0 then p = 2. Re-prove this using Mahler's theorem, and this time show conversely that a unique such f exists when p = 2.

You'll note that these are the same finite differences that one uses on polynomials in high school math contests, which is why they are also called "Mahler differences".

a 0 = f (0) a 1 = f (1) -f (0) a 2 = f (2) -2f (1) + f (0) a 3 = f (3) -3f (2) + 3f (1) -f (0).

Thus one can think of a n → 0 as saying that the values of f (0), f (1), . Let f : Z p → Q p be analytic. Then f has finitely many zeros.

To give an application of these results, we will prove the following result, which was interesting even before p-adics came along! Theorem 27.4.6 (Skolem-Mahler-Lech) Let (x i ) i≥0 be an integral linear recurrence, meaning (x i ) i≥0 is a sequence of integers

x n = c 1 x n-1 + c 2 x n-2 + • • • + c k x n-k n = 1, 2, . . .

holds for some choice of integers c 1 , . . . , c k . Then the set of indices {i | x i = 0} is eventually periodic.

Proof. According to the theory of linear recurrences, there exists a matrix A such that we can write x i as a dot product

x i = A i u, v .

Let p be a prime not dividing det A. Let T be an integer such that A T ≡ id (mod p) (with id denoting the identity matrix). Fix any 0 ≤ r < N . We will prove that either all the terms

f (n) = x nT +r n = 0, 1, . . .

are zero, or at most finitely many of them are. This will conclude the proof.

Let A T = id + pB for some integer matrix B. We have

f (n) = A nT +r u, v = ⟨(id + pB) n A r u, v⟩ = k≥0 n k • p n ⟨B n A r u, v⟩ = k≥0 a n n k where a n = p n ⟨B n A r u, v⟩ ∈ p n Z.

Thus we have written f in Mahler form. Initially, we define f : Z ≥0 → Z, but by Mahler's theorem (since lim n a n = 0) it follows that f extends to a function f : Z p → Q p . Also, we can check that lim n an n! = 0 hence f is even analytic. Thus by Strassman's theorem, f is either identically zero, or else it has finitely many zeros, as desired. §27.5 A few harder problems to think about Problem 27A † (Z p is compact). Show that Q p is not compact, but Z p is. (For the latter, I recommend using sequential continuity.) Problem 27B † (Totally disconnected). Show that both Z p and Q p are totally disconnected: there are no connected sets other than the empty set and singleton sets.

Problem 27C (Mentioned in MathOverflow). Let p be a prime. Find a sequence q 1 , q 2 , . . . of rational numbers such that:

• the sequence q n converges to 0 in the real sense;

• the sequence q n converges to 2021 in the p-adic sense.

Problem 27D (USA TST 2011). Let p be a prime. We say that a sequence of integers {z n } ∞ n=0 is a p-pod if for each e ≥ 0, there is an N ≥ 0 such that whenever m ≥ N , p e divides the sum

m k=0 (-1) k m k z k .

Prove that if both sequences {x n } ∞ n=0 and {y n } ∞ n=0 are p-pods, then the sequence Here is the picture. Suppose f : R → R is differentiable (hence continuous). We draw a graph of f in the usual way and consider values of h. For any nonzero h, what we get is the slope of the secant line joining (p, f (p)) to (p + h, f (p + h)). However, as h gets close to zero, that secant line begins to approach a line which is tangent to the graph of the curve. A picture with f a parabola is shown below, with the tangent in red, and the secant in dashed green.

{x n y n } ∞ n=0 is a p-pod.

y x (p, f (p)) Slope f ′ (p) (p + h, f (p + h)) Slope f (p+h)-f (p) h

So the picture in your head should be that f ′ (p) looks like the slope of the tangent line at (p, f (p)). 1 We will almost always use U = (a, b) or U = R, and you will not lose much by restricting the definition to those. 2 Remember we are following the convention in Abuse of Notation 26.6.6. So we mean "the limit of the function h → f (p+h)-f (p) h except the value at h = 0 can be anything". And this is important because that fraction does not have a definition at h = 0. As promised, we pay this no attention. 

f (p + h) -f (p) h = (p + h) 3 -p 3 h = 3p 2 h + 3ph 2 + h 3 h = 3p 2 + 3ph + h 2 . Thus, lim h→0 f (p + h) -f (p) h = lim h→0 (3p 2 + 3ph + h 2 ) = 3p 2 .

Thus the slope at each point of f is given by the formula 3p 2 . It is customary to then write f ′ (x) = 3x 2 as the derivative of the entire function f .

Abuse of Notation 28.1.5. We will now be sloppy and write this as (x 3 ) ′ = 3x 2 . This is shorthand for the significantly more verbose "the real-valued function x 3 on domain so-and-so has derivative 3p 2 at every point p in its domain".

In general, a real-valued differentiable function f : U → R naturally gives rise to derivative f ′ (p) at every point p ∈ U , so it is customary to just give up on p altogether and treat f ′ as function itself U → R, even though this real number is of a "different interpretation": f ′ (p) is meant to interpret a slope (e.g. your hourly pay rate) as opposed to a value (e.g. your total dollar worth at time t). If f is a function from real life, the units do not even match! This convention is so deeply entrenched I cannot uproot it without more confusion than it is worth. But if you read the chapters on multivariable calculus you will see how it comes back to bite us, when I need to re-define the derivative to be a linear map, rather than single real numbers.

§28.2 How to compute them

Same old, right? Sum rule, all that jazz.

Theorem 28.2.1 (Your friendly high school calculus rules)

In what follows f and g are differentiable functions, and U , V are open subsets of R.

• (Sum rule) If f, g : U → R then then (f + g) ′ (x) = f ′ (x) + g ′ (x). • (Product rule) If f, g : U → R then then (f • g) ′ (x) = f ′ (x)g(x) + f (x)g ′ (x). • (Chain rule) If f : U → V and g : V → R then the derivative of the composed function g • f : U → R is g ′ (f (x)) • f ′ (x).

Proof.

• Sum rule: trivial, do it yourself if you care.

• Product rule: for every nonzero h and point p ∈ U we may write

f (p + h)g(p + h) -f (p)g(p) h = f (p + h) -f (p) h • g(p + h) + g(p + h) -g(p) h • f (p)

which as h → 0 gives the desired expression.

• Chain rule: this is where Abuse of Notation 26.6.6 will actually bite us. Let p ∈ U , q = f (p) ∈ V , so that

(g • f ) ′ (p) = lim h→0 g(f (p + h)) -g(q) h .

We would like to write the expression in the limit as

g(f (p + h)) -g(q) h = g(f (p + h)) -g(q) f (p + h) -q • f (p + h) -f (p) h .

The problem is that the denominator f (p + h) -f (p) might be zero. So instead, we define the expression

Q(y) = g(y)-g(q) y-q if y ̸ = q g ′ (q) if y = q

which is continuous since g was differentiable at q. Then, we do have the equality

g(f (p + h)) -g(q) h = Q (f (p + h)) • f (p + h) -f (p) h .

because if f (p + h) = q with h ̸ = 0, then both sides are equal to zero anyways.

Then, in the limit as h → 0, we have lim h→0

f (p+h)-f (p) h = f ′ (p), while lim h→0 Q(f (p+ h)) = Q(q) = g ′ (

q) by continuity. This was the desired result. Remark 28.2.3 -Quick linguistic point: the theorems above all hold at each individual point. For example the sum rule really should say that if f, g : U → R are differentiable at the point p then so is f + g and the derivative equals f ′ (p) + g ′ (p). Thus if f and g are differentiable on all of U , then it of course follows that (f + g) ′ = f ′ + g ′ . So each of the above rules has a "point-by-point" form which then implies the "whole U " form.

We only state the latter since that is what is used in practice. However, in the rare situations where you have a function differentiable only at certain points of U rather than the whole interval U , you can still use the below.

We next list some derivatives of well-known functions, but as we do not give rigorous definitions of these functions, we do not prove these here.

Proposition 28.2.4 (Derivatives of some well-known functions)

• The exponential function exp : R → R defined by exp(x) = e x is its own derivative.

• The trig functions sin and cos have sin ′ = cos, cos ′ = -sin.

Example 28.2.5 (A typical high-school calculus question)

This means that you can mechanically compute the derivatives of any artificial function obtained by using the above, which makes it a great source of busy work in American high schools and universities. For example, if

f (x) = e x + x sin(x 2 ) f : R → R

then one can compute f ′ by:

f ′ (x) = (e x ) ′ + (x sin(x 2 )) ′ sum rule = e x + (x sin(x 2 )) ′ above table = e x + (x) ′ sin(x 2 ) + x(sin(x 2 )) ′ product rule = e x + sin(x 2 ) + x(sin(x 2 )) ′ (x) ′ = 1 = e x + sin(x 2 ) + x • 2x • cos(x 2 ) chain rule.

Of course, this function f is totally artificial and has no meaning, which is why calculus is the topic of widespread scorn in the United States. That said, it is worth appreciating that calculations like this are possible: one could say we have a pseudo-theorem "derivatives can actually be computed in practice".

If we take for granted that (e x ) ′ = e x , then we can derive two more useful functions to add to our library of functions we can differentiate.

Corollary 28.2.6 (Power rule)

Let r be a real number. The function R >0 → R by x → x r has derivative (x r ) ′ = rx r-1 .

Proof. We knew this for integers r already, but now we can prove it for any positive real number r. Write

f (x) = x r = e r log x

considered as a function f : R >0 → R. The chain rule (together with the fact that (e x ) ′ = e x ) now gives

f ′ (x) = e r log x • (r log x) ′ = e r log x • r x = x r • r x = rx r-1 .

The reason we don't prove the formulas for e x and log x is that we don't at the moment even have a rigorous definition for either, or even for 2 x if x is not rational. However it's nice to know that some things imply the other. The function log : R >0 → R has derivative (log x) ′ = 1/x.

Proof. We have that x = e log x . Differentiate both sides, and again use the chain rule3 1 = e log x • (log x) ′ .

Thus (log x) ′ = 1 e log x = 1/x.

§28.3 Local (and global) maximums

Prototypical example for this section: Horizontal tangent lines to the parabola are typically good pictures.

You may remember from high school that one classical use of calculus was to extract the minimum or maximum values of functions. We will give a rigorous description of how to do this here.

Definition 28.3.1. Let f : U → R be a function. A local maximum is a point p ∈ U such that there exists an open neighborhood V of p (contained inside U ) such that f (p) ≥ f (x) for every x ∈ V .

A local minimum is defined similarly.4 

Definition 28.3.2. A point p is a local extrema if it satisfies either of these.

The nice thing about derivatives is that they pick up all extrema.

Theorem 28.3.3 (Fermat's theorem on stationary points)

Suppose f : U → R is differentiable and p ∈ U is a local extrema. Then f ′ (p) = 0.

If you draw a picture, this result is not surprising. (Note also: the converse is not true. Say, f (x) = x 2019 has f ′ (0) = 0 but x = 0 is not a local extrema for f .)

Proof. Assume for contradiction f ′ (p) > 0. Choose any ε > 0 with ε < f ′ (p). Then for sufficiently small |h| we should have

f (p + h) -f (p) h > ε.

In particular

f (p + h) > f (p) for h > 0 while f (p -h) < f (p) for h < 0. So p is not a local extremum.

The proof for f ′ (p) < 0 is similar.

However, this is not actually adequate if we want a complete method for optimization. The issue is that we seek global extrema, which may not even exist: for example f (x) = x (which has f ′ (x) = 1) obviously has no local extrema at all. The key to resolving this is to use compactness: we change the domain to be a compact set Z, for which we know that f will achieve some global maximum. The set Z will naturally have some interior S, and calculus will give us all the extrema within S. Then we manually check all cases outside Z.

Let's see two extended examples. The one is simple, and you probably already know about it, but I want to show you how to use compactness to argue thoroughly, and how the "boundary" points naturally show up.

Example 28.3.4 (Rectangle area optimization)

Suppose we consider rectangles with perimeter 20 and want the rectangle with the smallest or largest area.

10x x

If we choose the legs of the rectangle to be x and 10 -x, then we are trying to optimize the function

f (x) = x(10 -x) = 10x -x 2 f : [0, 10] → R.

By compactness, there exists some global maximum and some global minimum.

As f is differentiable on (0, 10), we find that for any p ∈ (0, 10), a global maximum will be a local maximum too, and hence should satisfy

0 = f ′ (p) = 10 -2p =⇒ p = 5.

Also, the points x = 0 and x = 10 lie in the domain but not the interior (0, 10). Therefore the global extrema (in addition to existing) must be among the three suspects {0, 5, 10}.

We finally check f (0) = 0, f (5) = 25, f (10) = 0. So the 5 × 5 square has the largest area and the degenerate rectangles have the smallest (zero) area.

Here is a non-elementary example. For all real numbers x we have e x ≥ 1 + x.

Proof. Define the differentiable function

f (x) = e x -(x + 1) f : R → R. Consider the compact interval Z = [-1, 100]. If x ≤ -1 then obviously f (x) > 0.

Similarly if x ≥ 100 then obviously f (x) > 0 too. So we just want to prove that if x ∈ Z, we have f (x) ≥ 0.

Indeed, there exists some global minimum p. It could be the endpoints -1 or 100. Otherwise, if it lies in U = (-1, 100) then it would have to satisfy

0 = f ′ (p) = e p -1 =⇒ p = 0.

As f (-1) > 0, f (100) > 0, f (0) = 0, we conclude p = 0 is the global minimum of Z; and hence f (x) ≥ 0 for all x ∈ Z, hence for all x.

Remark 28.3.6 -If you are willing to use limits at ±∞, you can rewrite proofs like the above in such a way that you don't have to explicitly come up with endpoints like -1 or 100. We won't do so here, but it's nice food for thought.

§28.4 Rolle and friends

Prototypical example for this section: The racetrack principle, perhaps?

One corollary of the work in the previous section is Rolle's theorem.

Theorem 28.4.1 (Rolle's theorem) Image from [Mu] One can adapt the theorem as follows. 

Suppose f : [a, b] → R is a continuous function, which is differentiable on the open interval (a, b), such that f (a) = f (b). Then there is a point c ∈ (a, b) such that f ′ (c) = 0.

f ′ (c) = f (b) -f (a) b -a .

Pictorially, there is a c such that the tangent at c has the same slope as the secant joining (a, f (a)), to (b, f (b)); and Rolle's theorem is the special case where that secant is horizontal.

y x (a, f (a)) (b, f (b)) Slope f (b)-f (a) b-a a b (c, f (c)) Proof of mean value theorem. Let s = f (b)-f (a)

b-a be the slope of the secant line, and define g(x) = f (x) -sx which intuitively shears f downwards so that the secant becomes vertical. In fact g(a) = g(b) now, so we apply Rolle's theorem to g. Remark 28.4.3 (For people with driver's licenses) -There is a nice real-life interpretation of this I should mention. A car is travelling along a one-dimensional road (with f (t) denoting the position at time t). Suppose you cover 900 kilometers in your car over the course of 5 hours (say f (0) = 0, f (5) = 900). Then there is some point at time in which your speed at that moment was exactly 180 kilometers per hour, and so you cannot really complain when the cops pull you over for speeding.

The mean value theorem is important because it lets you relate use derivative information to get information about the function in a way that is really not possible without it. Here is one quick application to illustrate my point:

Proposition 28.4.4 (Racetrack principle) Let f, g : R → R be two differentiable functions with f (0) = g(0). (a) If f ′ (x) ≥ g ′ (x) for every x > 0, then f (x) ≥ g(x) for every x > 0. (b) If f ′ (x) > g ′ (x) for every x > 0, then f (x) > g(x) for every x > 0.

This proposition might seem obvious. You can think of it as a race track for a reason: if f and g denote the positions of two cars (or horses etc) and the first car is always faster than the second car, then the first car should end up ahead of the second car. At a special case g = 0, this says that if f ′ (x) ≥ 0, i.e. "f is increasing", then, well, f (x) ≥ f (0) for x > 0, which had better be true. However, if you try to prove this by definition from derivatives, you will find that it is not easy! However, it's almost a prototype for the mean value theorem.

Proof of racetrack principle. We prove (a). Let h = f -g, so h(0) = 0. Assume for contradiction h(p) < 0 for some p > 0. Then the secant joining (0, h(0)) to (p, h(p)) has negative slope; in other words by mean value theorem there is a 0 < c < p such that

f ′ (c) -g ′ (c) = h ′ (c) = h(p) -h(0) p = h(p) p < 0 so f ′ (c) < g ′ (c), contradiction. Part (b) is the same.

Sometimes you will be faced with two functions which you cannot easily decouple; the following form may be more useful in that case. 

f ′ (c)(g(b) -g(a)) = g ′ (c)(f (b) -f (a))

Proof. Use Rolle's theorem on the function

h(x) = [f (x) -f (a)] [g(b) -g(a)] -[g(x) -g(a)] [f (b) -f (a)] .

This is also called Cauchy's mean value theorem or the extended mean value theorem. §28.5 Smooth functions Prototypical example for this section: All the functions you're used to.

Let f : U → R be differentiable, thus giving us a function f ′ : U → R. If our initial function was nice enough, then we can take the derivative again, giving a function f ′′ : U → R, and so on. In general, after taking the derivative n times, we denote the resulting function by f (n) . By convention, f (0) = f . Definition 28.5.1. A function f : U → R is smooth if it is infinitely differentiable; that is the function f (n) exists for all n.

Question 28.5.2. Show that the absolute value function is not smooth.

Most of the functions we encounter, such as polynomials, e x , log, sin, cos are smooth, and so are their compositions. Here is a weird example which we'll grow more next time.

Example 28.5.3 (A smooth function with all derivatives zero)

Consider the function 

f (x) = e -1/x x > 0 0 x ≤ 0.

h ′ (x) = f ′ (x)g(x) -f (x)g ′ (x) g(x) 2 .

Problem 28B. For real numbers x > 0, how small can x x be?

Problem 28C (RMM 2018). Determine whether or not there exist nonconstant polynomials P (x) and Q(x) with real coefficients satisfying 20 .

P (x) 10 + P (x) 9 = Q(x) 21 + Q(x)

Problem 28D. Let P (x) be a degree n polynomial with real coefficients. Prove that the equation e x = P (x) has at most n + 1 real solutions in x.

Problem 28E (Jensen's inequality). Let f : (a, b) → R be a twice differentiable function such that f ′′ (x) ≥ 0 for all x (i.e. f is convex). Prove that

f x + y 2 ≤ f (x) + f (y) 2

for all real numbers x and y in the interval (a, b).

Problem 28F (L'Hôpital rule, or at least one case). Let f, g : R → R be differentiable functions and let p be a real number. Suppose that

lim x→p f (x) = lim x→p g(x) = 0.

Prove that

lim x→p f (x) g(x) = lim x→p f ′ (x) g ′ (x)

provided the right-hand limit exists.

Polynomials are very well-behaved functions, and are studied extensively for that reason. From an analytic perspective, for example, they are smooth, and their derivatives are easy to compute.

In this chapter we will study power series, which are literally "infinite polynomials" n a n x n . Armed with our understanding of series and differentiation, we will see three great things:

• Many of the functions we see in nature actually are given by power series. Among them are e x , log x, sin x.

• Their convergence properties are actually quite well behaved: from the string of coefficients, we can figure out which x they converge for.

• The derivative of n a n x n is actually just n na n x n-1 .

§29.1 Motivation

To get the ball rolling, let's start with one infinite polynomial you'll recognize: for any fixed number -1 < x < 1 we have the series convergence

1 1 -x = 1 + x + x 2 + . . .

by the geometric series formula.

Let's pretend we didn't see this already in Problem 26D. So, we instead have a smooth function f : (-1, 1) → R by

f (x) = 1 1 -x .

Suppose we wanted to pretend that it was equal to an "infinite polynomial" near the origin, that is

(1 -x) -1 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + . . . .

How could we find that polynomial, if we didn't already know? Well, for starters we can first note that by plugging in x = 0 we obviously want a 0 = 1.

We have derivatives, so actually, we can then differentiate both sides to obtain that

(1 -x) -2 = a 1 + 2a 2 x + 3a 3 x 2 + 4a 4 x 3 + • • • .

If we now set x = 0, we get a 1 = 1. In fact, let's keep taking derivatives and see what we get.

(

-x) -1 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 + . . . (1 -x) -2 = a 1 + 2a 2 x + 3a 3 x 2 + 4a 4 x 3 + 5a 5 x 4 + . . . 2(1 -x) -3 = 2a 2 + 6a 3 x + 12a 4 x 2 + 20a 5 x 3 + . . . 6(1 -x) -4 = 6a 3 + 24a 4 x + 60a 5 x 2 + . . . 24(1 -x) -5 = 24a 4 + 120a 5 x + . . . . . . .If we set x = 0 we find 1 = a 0 = a 1 = a 2 = . . . which is what we expect; the geometric series 1 1-x = 1 + x + x 2 + . . . . And so actually taking derivatives was enough to get the right claim! §29.2 Power series Prototypical example for this section: 1 1-z = 1 + z + z 2 + . . . , which converges on (-1, 1). Of course this is not rigorous, since we haven't described what the right-hand side is, much less show that it can be differentiated term by term. So we define the main character now.

Definition 29.2.1.

A power series is a sum of the form

∞ n=0 a n z n = a 0 + a 1 z + a 2 z 2 + . . .

where a 0 , a 1 , . . . are real numbers, and z is a variable.

Abuse of Notation 29.2.2 (0 0 = 1). If you are very careful, you might notice that when z = 0 and n = 0 we find 0 0 terms appearing. For this chapter the convention is that they are all equal to one. Now, if I plug in a particular real number h, then I get a series of real numbers ∞ n=0 a n h n . So I can ask, when does this series converge? It terms out there is a precise answer for this. Let ∞ n=0 a n z n be a power series with radius of convergence R. Let h be a real number, and consider the infinite series ∞ n=0 a n h n of real numbers. Then:

• The series converges absolutely if |h| < R.

• The series diverges if |h| > R.

Proof. This is not actually hard, but it won't be essential, so not included. Therefore, if n a n z n is a power series with a nonzero radius R > 0 of convergence, then it can also be thought of as a function

(-R, R) → R by h → n≥0 a n h n .

This is great. Note also that if R = ∞, this means we get a function R → R.

Abuse of Notation 29.2.7 (Power series vs. functions).

There is some subtlety going on with "types" of objects again. Analogies with polynomials can help.

Consider P (x) = x 3 + 7x + 9, a polynomial. You can, for any real number h, plug in P (h) to get a real number. However, in the polynomial itself, the symbol x is supposed to be a variable -which sometimes we will plug in a real number for, but that happens only after the polynomial is defined.

Despite this, "the polynomial p(x) = x 3 + 7x + 9" (which can be thought of as the coefficients) and "the real-valued function x → x 3 +7x+9" are often used interchangeably. The same is about to happen with power series: while they were initially thought of as a sequence of coefficients, the Cauchy-Hadamard theorem lets us think of them as functions too, and thus we blur the distinction between them.

§29.3 Differentiating them

Prototypical example for this section: We saw earlier 1 + x + x 2 + x 3 + . . . has derivative 1 + 2x + 3x 2 + . . . . As promised, differentiation works exactly as you want.

Theorem 29.3.1 (Differentiation works term by term)

Let n≥0 a n z n be a power series with radius of convergence R > 0, and consider the corresponding function

f : (-R, R) → R by f (x) = n≥0 a n x n .

Then all the derivatives of f exist and are given by power series

f ′ (x) = n≥1 na n x n-1 f ′′ (x) = n≥2 n(n -1)a n x n-2 . . .

which also converge for any x ∈ (-R, R). In particular, f is smooth.

Proof. Also omitted. The right way to prove it is to define the notion "converges uniformly", and strengthen Cauchy-Hadamard to have this as a conclusion as well. However, we won't use this later.

Corollary 29.3.2 (A description of power series coefficients)

Let n≥0 a n z n be a power series with radius of convergence R > 0, and consider the corresponding function f (x) as above. Then

a n = f (n) (x) n! .

Proof. Take the nth derivative and plug in x = 0.

§29.4 Analytic functions

Prototypical example for this section: The piecewise e -1/x or 0 function is not analytic, but is smooth.

With all these nice results about power series, we now have a way to do this process the other way: suppose that f : U → R is a function. Can we express it as a power series?

Functions for which this is true are called analytic. Moreover, if f is analytic, then by the corollary above its coefficients are actually described exactly by

f (x) = n≥0 f (n) (p) n! (x -p) n .

Even if f is smooth but not analytic, we can at least write down the power series; we give this a name. 

f (x) = exp(-1/x) x > 0 0 x ≤ 0

is not analytic. Indeed, suppose for contradiction it was. As all the derivatives are zero, its Taylor series would be 0 + 0x + 0x 2 + . . . . This Taylor series does converge, but not to the right value -as f (ε) > 0 for any ε > 0, contradiction. It now follows that f (x) = sin(x) is analytic. To see that, we can compute

f (0) = sin 0 = 0 f ′ (0) = cos 0 = 1 f ′′ (0) = -sin 0 = 0 f (3) (0) = -cos 0 = -1 f (4) (0) = sin 0 = 0 f (5) (0) = cos 0 = 1 f (6) (0) = -sin 0 = 0 . . .

and so by continuing the pattern (which repeats every four) we find the Taylor series is

z - z 3 3! + z 5 5! - z 7 7! + . . .

which is seen to have radius of convergence R = ∞.

Like with differentiable functions:

Proposition 29.4.7 (All your usual closure properties for analytic functions)

The sums, products, compositions, nonzero quotients of analytic functions are analytic.

The upshot of this is that most of your usual functions that occur in nature, or even artificial ones like f (x) = e x + x sin(x 2 ), will be analytic, hence describable locally by Taylor series.

§29.5 A definition of Euler's constant and exponentiation

We can actually give a definition of e x using the tools we have now.

Definition 29.5.1. We define the map exp : R → R by using the following power series, which has infinite radius of convergence:

exp(x) = n≥0 x n n! .

We then define Euler's constant as e = exp(1). We are then settled with:

Proposition 29.

5.3 (exp is multiplicative)

Under this definition, exp(x + y) = exp(x) exp(y).

Idea of proof.

There is some subtlety here with switching the order of summation that we won't address. Modulo that: The log function then comes after.

exp(x) exp(y) = n≥0 x n n! m≥0 y m m! = n≥0 m≥0 x n n! y m m! = k≥0 m+n=k m,n≥0 x n y m n!m! = k≥0 m+n=k m,n≥0 k n x n y m k! = k≥0 (x + y) k k! = exp(x + y).

Definition 29.5.5. We may define log : R >0 → R to be the inverse function of exp.

Since its derivative is 1/x it is smooth; and then one may compute its coefficients to show it is analytic.

Note that this actually gives us a rigorous way to define a r for any a > 0 and r > 0, namely a r := exp (r log a) . §29.6 This all works over complex numbers as well, except also complex analysis is heaven

We now mention that every theorem we referred to above holds equally well if we work over C, with essentially no modifications.

• Power series are defined by n a n z n with a n ∈ C, rather than a n ∈ R.

• The definition of radius of convergence R is unchanged! The series will converge if |z| < R.

• Differentiation still works great. (The definition of the derivative is unchanged.)

• Analytic still works great for functions f :

U → C, with U ⊆ C open.

In particular, we can now even define complex exponentials, giving us a function (We still require the base a to be a positive real so that log a is defined, though. So this i i issue is still there.) However, if one tries to study calculus for complex functions as we did for the real case, in addition to most results carrying over, we run into a huge surprise:

If f : C → C is differentiable, it is analytic.

And this is just the beginning of the nearly unbelievable results that hold for complex analytic functions. But this is the part on real analysis, so you will have to read about this later! §29. 

P n = n k=0 f (k) (a) k! • (b -a) k .

Prove that there exists ξ ∈ (a, b) such that

f (n+1) (ξ) = (n + 1)! • f (b) -P n (b -a) n+1 .

This generalizes the mean value theorem (which is the special case n = 0, where

P 0 = f (a)).

Problem 29D (Putnam 2018 A5). Let f : R → R be smooth, and assume that f (0) = 0, f (1) = 1, and f (x) ≥ 0 for every real number x. Prove that f (n) (x) < 0 for some positive integer n and real number x.

"Trying to Riemann integrate discontinuous functions is kind of outdated." -Dennis Gaitsgory, [Ga15] We will go ahead and define the Riemann integral, but we won't do very much with it. The reason is that the Lebesgue integral is basically better, so we will define it, check the fundamental theorem of calculus (or rather, leave it as a problem at the end of the chapter), and then always use Lebesgue integrals forever after. §30.1 Uniform continuity Prototypical example for this section: f (x) = x 2 is not uniformly continuous on R, but functions on compact sets are always uniformly continuous. Definition 30.1.1. Let f : M → N be a continuous map between two metric spaces. We say that f is uniformly continuous if for all ε > 0 there exists a δ > 0 such that

d M (p, q) < δ =⇒ d N (f (p), f (q)) < ε.

This difference is that given an ε > 0 we must specify a δ > 0 which works for every choice p and q of inputs; whereas usually δ is allowed to depend on p and q. (Also, this definition can't be ported to a general topological space.) Example 30.1.2 (Uniform continuity failure) (a) The function f : R → R by x → x 2 is not uniformly continuous. Suppose we take ε = 0.1 for example. There is no δ such that if |x -y| < δ then |x 2 -y 2 | < 0.1, since as x and y get large, the function f becomes increasingly sensitive to small changes.

(b) The function (0, 1) → R by x → x -1 is not uniformly continuous.

(c) The function R >0 → R by x → √ x does turn out to be uniformly continuous (despite having unbounded derivatives!). Indeed, you can check that the assertion

|x -y| < ε 2 =⇒ √ x - √ y < ε holds for any x, y, ε > 0.

The good news is that in the compact case all is well. Let M be a compact metric space. Then any continuous map f : M → N is also uniformly continuous.

Proof. Assume for contradiction there is some bad ε > 0. Then taking δ = 1/n, we find that for each integer n there exists points p n and q n which are within 1/n of each other, but are mapped more than ε away from each other by f . In symbols,

d M (p n , q n ) < 1/n but d N (f (p n ), f (q n )) > ε.

By compactness of M , we can find a convergent subsequence p i 1 , p i 2 , . . . converging to some x ∈ M . Since the q in is within 1/i n of p in , it ought to converge as well, to the same point x ∈ M . Then the sequences f (p in ) and f (q in ) should both converge to f (x) ∈ N , but this is impossible as they are always ε away from each other. 

(a) Q is dense in R.

(b) In general, any metric space M is dense in its completion M .

Dense sets lend themselves to having functions completed. The idea is that if I have a continuous function f : Q → N , for some metric space N , then there should be at most one way to extend it to a function f : R → N . For we can approximate each rational number by real numbers: if I know f (1), f (1.4), f (1.41), . . . f ( √ 2) had better be the limit of this sequence. So it is certainly unique.

However, there are two ways this could go wrong:

Example 30.2.3 (Non-existence of extension) (a) It could be that N is not complete, so the limit may not even exist in N . For example if N = Q, then certainly there is no way to extend even the identity function f : Q → N to a function f : R → N .

(b) Even if N was complete, we might run into issues where f explodes. For example, let N = R and define

f (x) = 1 x - √ 2 f : Q → R.

There is also no way to extend this due to the explosion of f near √ 2 / ∈ Q, which would cause f ( √ 2) to be undefined.

However, the way to fix this is to require f to be uniformly continuous, and in that case we do get a unique extension. Hence we define ψ(x) to be the limit of that sequence; this doesn't depend on the choice of sequence, and one can use sequential continuity to show ψ is continuous.

§30.3 Defining the Riemann integral

Extensions will allow us to define the Riemann integral. I need to introduce a bit of notation so bear with me. 

= t 0 < t 1 < t 2 < • • • < t n = b. • We let M ([a, b]) = C 0 ([a, b]) ∪ R([a, b]).

Warning: only C 0 ([a, b]) is common notation, and the other two are made up.

See picture below for a typical a rectangle function. (It is irritating that we have to officially assign a single value to each t i , even though there are naturally two values we want to use, and so we use the convention of letting the left endpoint be closed).

a = t 0 b = t 4 t 1 t 2 t 3 Definition 30.3.2. We can impose a metric on M ([a, b]) by defining d(f, g) = sup x∈[a,b] |f (x) -g(x)| .

Now, there is a natural notion of integral for rectangle functions: just sum up the obvious rectangles! Officially, this is the expression

f (a)(t 1 -a) + f (t 1 )(t 2 -t 1 ) + +f (t 2 ) (t 3 -t 2 ) + • • • + f (t n ) (b -t n ) .

We denote this function by Σ : R([a, b]) → R.

Theorem 30.3.3 (The Riemann integral)

There exists a unique continuous map 

a : M ([a, b]) → R such that the diagram M ([a, b]) R R([a, b]) b a ⊃ Σ commutes.

Proof. We want to apply the extension theorem, so we just have to check a few things:

• We claim R([a, b]) is a dense subset of M ([a, b]).

In other words, for any continuous f : [a, b] → R and ε > 0, we want there to exist a rectangle function that approximates f within ε.

This follows by uniform continuity. We know there exists a δ > 0 such that whenever |x -y| < δ we have |f (x) -f (y)| < ε. So as long as we select a rectangle function whose rectangles have width less than δ, and such that the upper-left corner of each rectangle lies on the graph of f , then we are all set.

a b

• The "add-the-rectangles" map Σ : R([a, b]) → R is uniformly continuous. Actually this is pretty obvious: if two rectangle functions f and

g have d(f, g) < ε, then d(Σf, Σg) < ε(b -a).

• R is complete. 

a = t 0 < t 1 < t 2 < • • • < t n = b and ξ i ∈ [t i-1 , t i ] ∀ 1 ≤ i ≤ n.

The mesh of P is the width of the longest interval, i.e. max i (t i -t i-1 ).

Of course the point of this definition is that we add the rectangles, but the ξ i are the sample points. f (ξ i )(t i -t i-1 ) .

Here the limit means that we can take any sequence of partitions whose mesh approaches zero.

Proof. The right-hand side corresponds to the areas of some rectangle functions g 1 , g 2 , . . . with increasingly narrow rectangles. As in the proof Theorem 30.3.3 , as the meshes of those rectangles approaches zero, by uniform continuity, we have d(f, g n ) → 0 as well. Thus by continuity in the diagram of Theorem 30.3.3, we get lim n Σ(g n ) = (f ) as needed.

Combined with the mean value theorem, this can be used to give a short proof of the fundamental theorem of calculus for functions f with a continuous derivative. The idea is that for any choice of partition a ≤ t 0 < t 1 < t 2 < • • • < t n ≤ b, using the Mean Value Theorem it should be possible to pick ξ i in each interval to match with the slope of the secant: at which point the areas sum to the total change in f . We illustrate this situation with three points, and invite the reader to fill in the details as Problem 30B ⋆ .

y x f t 0 t 1 t 2 t 3 ξ 1 ξ 2 ξ 3

Net change

One quick note is that although I've only defined the Riemann integral for continuous functions, there ought to be other functions for which it exists (including "piecewise continuous functions" for example, or functions "continuous almost everywhere"). The relevant definition is:

Definition 30.4.3. If f : [a, b] → R

is a function which is not necessarily continuous, but for which the limit lim

P tagged partition mesh P →0 n i=1 f (ξ i )(t i -t i-1 ) .

exists anyways, then we say f is Riemann integrable on [a, b] and define its value to be that limit b a f (x) dx. We won't really use this definition much, because we will see that every Riemann integrable function is Lebesgue integrable, and the Lebesgue integral is better.

Example 30.4.4 (Your AP calculus returns)

We had better mention that Problem 30B ⋆ implies that we can compute Riemann integrals in practice, although most of you may already know this from high-school calculus. For example, on the interval (1, 4), the derivative of the function

F (x) = 1 3 x 3 is F ′ (x) = x 2 . As f (x) = x 2 is a continuous function f : [1, 4] → R, we get 4 1 x 2 dx = F (4) -F (1) = 64 3 - 1 3 = 21.

Note that we could also have picked F (x) = 1 3 x 3 + 2019; the function F is unique up to shifting, and this constant cancels out when we subtract. This is why it's common in high school to (really) abuse notation and write x 2 dx = 

′ : [a, b] → R. Prove that b a f ′ (x) dx = f (b) -f (a).

Problem 30C ⋆ (Improper integrals). For each real number r > 0, evaluate the limit

1 lim ε→0 + 1 ε 1 x r dx

or show it does not exist. This can intuitively be thought of as the "improper" integral1 0 x -r dx; it doesn't make sense in our original definition since we did not (and cannot) define the integral over the non-compact (0, 1] but we can still consider the integral over [ε, 1] for any ε > 0.

Problem 30D. Show that lim n→∞ 1 n + 1 + 1 n + 2 + • • • + 1 2n = log 2.

IX Complex Analysis

Part IX: Contents 31 Holomorphic functions 31. 1 The nicest functions on earth . 

32

Meromorphic functions 32. 1 The second nicest functions on earth . 

33

Holomorphic square roots and logarithms 33.1 Motivation: square root of a complex number . 

34

Bonus: Topological Abel-Ruffini Theorem 34.1 The Game Plan . 

34.3

Step 2: Nested Roots . . . 34.4 Step 3: Normal Groups . In high school you were told how to differentiate and integrate real-valued functions. In this chapter on complex analysis, we'll extend it to differentiation and integration of complex-valued functions.

Big deal, you say. Calculus was boring enough. Why do I care about complex calculus? Perhaps it's easiest to motivate things if I compare real analysis to complex analysis. In real analysis, your input lives inside the real line R. This line is not terribly discerningyou can construct a lot of unfortunate functions. Here are some examples. is continuous everywhere but differentiable nowhere.

(c) The function x → x 100 x ≥ 0 -x 100 x < 0 has the first 99 derivatives but not the 100th one.

(d) If a function has all derivatives (we call these smooth functions), then it has a Taylor series. But for real functions that Taylor series might still be wrong. The function

x → e -1/x x > 0 0

x ≤ 0 has derivatives at every point. But if you expand the Taylor series at x = 0, you get 0 + 0x + 0x 2 + . . . , which is wrong for any x > 0 (even x = 0.0001).

Let's even put aside the pathology. If I tell you the value of a real smooth function on the interval [-1, 1], that still doesn't tell you anything about the function as a whole. It could be literally anything, because it's somehow possible to "fuse together" smooth functions. So what about complex functions? If you consider them as functions R 2 → R 2 , you now have the interesting property that you can integrate along things that are not line segments: you can write integrals across curves in the plane. But C has something more: it is a field, so you can multiply and divide two complex numbers.

So we restrict our attention to differentiable functions called holomorphic functions. It turns out that the multiplication on C makes all the difference. The primary theme in what follows is that holomorphic functions are really, really nice, and that knowing tiny amounts of data about the function can determine all its values.

The two main highlights of this chapter, from which all other results are more or less corollaries:

• Contour integrals of loops are always zero.

• A holomorphic function is essentially given by its Taylor series; in particular, singledifferentiable implies infinitely differentiable. Thus, holomorphic functions behave quite like polynomials.

Some of the resulting corollaries:

• It'll turn out that knowing the values of a holomorphic function on the boundary of the unit circle will tell you the values in its interior.

• Knowing the values of the function at 1, 1 2 , 1 3 , . As [Pu02] writes: "Complex analysis is the good twin and real analysis is the evil one: beautiful formulas and elegant theorems seem to blossom spontaneously in the complex domain, while toil and pathology rule the reals".

§31.2 Complex differentiation

Prototypical example for this section: Polynomials are holomorphic; z is not.

Let f : U → C be a complex function. Then for some z 0 ∈ U , we define the derivative at z 0 to be

lim h→0 f (z 0 + h) -f (z 0 ) h .

Note that this limit may not exist; when it does we say f is differentiable at z 0 . What do I mean by a "complex" limit h → 0? It's what you might expect: for every ε > 0 there should be a δ > 0 such that

0 < |h| < δ =⇒ f (z 0 + h) -f (z 0 ) h -L < ε.

If you like topology, you are encouraged to think of this in terms of open neighborhoods in the complex plane. (This is why we require U to be open: it makes it possible to take δ-neighborhoods in it.) But note that having a complex derivative is actually much stronger than a real function having a derivative. In the real line, h can only approach zero from below and above, and for the limit to exist we need the "left limit" to equal the "right limit". But the complex numbers form a plane: h can approach zero from many directions, and we need all the limits to be equal.

Example 31.2.1 (Important: conjugation is not holomorphic) Let f (z) = z be complex conjugation, f : C → C. This function, despite its simple nature, is not holomorphic! Indeed, at z = 0 we have,

f (h) -f (0) h = h h .

This does not have a limit as h → 0, because depending on "which direction" we approach zero from we have different values.

Im

Re 0

1 1 i -1 f (z) = z f (0 + h) -f (0) h

If a function f : U → C is complex differentiable at all the points in its domain it is called holomorphic. In the special case of a holomorphic function with domain U = C, we call the function entire. 1

Example 31.2.2 (Examples of holomorphic functions)

In all the examples below, the derivative of the function is the same as in their real analogues (e.g. the derivative of e z is e z ).

(a) Any polynomial

z → z n + c n-1 z n-1 + • • • + c 0 is holomorphic. (b)

The complex exponential exp : x + yi → e x (cos y + i sin y) can be shown to be holomorphic.

(c) sin and cos are holomorphic when extended to the complex plane by cos z = e iz +e -iz 2

and sin z = e iz -e -iz 2i .

(d) As usual, the sum, product, chain rules and so on apply, and hence sums, products, nonzero quotients, and compositions of holomorphic functions are also holomorphic.

You are welcome to try and prove these results, but I won't bother to do so.

§31.3 Contour integrals

Prototypical example for this section: γ z m dz around the unit circle.

In the real line we knew how to integrate a function across a line segment [a, b]: essentially, we'd "follow along" the line segment adding up the values of f we see to get some area. Unlike in the real line, in the complex plane we have the power to integrate over arbitrary paths: for example, we might compute an integral around a unit circle. A contour integral lets us formalize this.

First of all, if f : R → C and f (t) = u(t) + iv(t) for u, v ∈ R, we can define an integral b a by just adding the real and imaginary parts:

b a f (t) dt = b a u(t) dt + i b a v(t) dt .

Now let α : [a, b] → C be a path, thought of as a complex differentiable2 function. Such a path is called a contour, and we define its contour integral by

α f (z) dz = b a f (α(t)) • α ′ (t) dt.

You can almost think of this as a u-substitution (which is where the α ′ comes from).

In particular, it turns out this integral does not depend on how α is "parametrized": a circle given by Then for any integer m, we have

[0, 2π] → C : t → e it

γ z m dz = 2πi m = -1 0 otherwise

Proof. The derivative of e it is ie it . So, by definition the answer is the value of

2π 0 (e it ) m • (ie it ) dt = 2π 0 i(e it ) 1+m dt = i 2π 0 cos[(1 + m)t] + i sin[(1 + m)t] dt = - 2π 0 sin[(1 + m)t] dt + i 2π 0 cos[(1 + m)t] dt.

This is now an elementary calculus question. One can see that this equals 2πi if m = -1 and otherwise the integrals vanish.

Let me try to explain why this intuitively ought to be true for m = 0. In that case we have γ 1 dz. So as the integral walks around the unit circle, it "sums up" all the tangent vectors at every point (that's the direction it's walking in), multiplied by 1. And given the nice symmetry of the circle, it should come as no surprise that everything cancels out. The theorem says that even if we multiply by z m for m ̸ = -1, we get the same cancellation.

Definition 31.3.2. Given α : [0, 1] → C, we denote by α the "backwards" contour α(t) = α(1 -t). This might seem a little boring. Things will get really cool really soon, I promise.

§31.4 Cauchy-Goursat theorem

Prototypical example for this section: γ z m dz = 0 for m ≥ 0. But if m < 0, Cauchy's theorem does not apply.

Let Ω ⊆ C be simply connected (for example, Ω = C), and consider two paths α, β with the same start and end points.

Ω α β

What's the relation between α f (z) dz and β f (z) dz? You might expect there to be some relation between them, considering that the space Ω is simply connected. But you probably wouldn't expect there to be much of a relation.

As a concrete example, let Ψ : C → C be the function z → z -Re[z] (for example, Ψ(2015 + 3i) = 3i). Let's consider two paths from -1 to 1. Thus β is walking along the real axis, and α which follows an upper semicircle.

- 

Ψ(z) dz = - α Ψ(z) dz = - π 0 (i sin(t)) • ie it dt = 1 2 πi

which in particular is not zero. But somehow Ψ is not a really natural function. It's not respecting any of the nice, multiplicative structure of C since it just rudely lops off the real part of its inputs. More precisely, Question 31.4.1. Show that Ψ(z) = z -Re[z] is not holomorphic. (Hint: z is not holomorphic.) Now here's a miracle: for holomorphic functions, the two integrals are always equal. Equivalently, (by considering α followed by β) contour integrals of loops are always zero. This is the celebrated Cauchy-Goursat theorem (also called the Cauchy integral theorem, but later we'll have a "Cauchy Integral Formula" so blah). z is holomorphic! The defect now is that Ω = C \ {0} is not simply connected. So the theorem passes our sanity checks, albeit barely.

The typical proof of Cauchy's Theorem assumes additionally that the partial derivatives of f are continuous and then applies the so-called Green's theorem. But it was Goursat who successfully proved the fully general theorem we've stated above, which assumed only that f was holomorphic. I'll only outline the proof, and very briefly. You can show that if f : Ω → C has an antiderivative F : Ω → C which is also holomorphic, and moreover Ω is simply connected, then you get a "fundamental theorem of calculus", a la

α f (z) dz = F (α(b)) -F (α(a))

where α : [a, b] → C is some path. So to prove Cauchy-Goursat, you only have to show this antiderivative F exists. Goursat works very hard to prove the result in the special case that γ is a triangle, and hence by induction for any polygon. Once he has the result for a rectangle, he uses this special case to construct the function F explicitly. Goursat then shows that F is holomorphic, completing the proof.

Anyways, the theorem implies that γ z m dz = 0 when m ≥ 0. So much for all our hard work earlier. But so far we've only played with circles. This theorem holds for any contour which is a loop. So what else can we do? §31.

5 Cauchy's integral theorem

We now present a stunning application of Cauchy-Goursat, a "representation theorem": essentially, it says that values of f inside a disk are determined by just the values on the boundary! In fact, we even write down the exact formula. As [Ya12] says, "any time a certain type of function satisfies some sort of representation theorem, it is likely that many more deep theorems will follow." Let's pull back the curtain: Theorem 31.5.1 (Cauchy's integral formula)

Let γ : [0, 2π] → C be a circle in the plane given by t → Re it , which bounds a disk D. Suppose f : U → C is holomorphic such that U contains the circle and its interior. Then for any point a in the interior of D, we have

f (a) = 1 2πi γ f (z) z -a dz.

Note that we don't require U to be simply connected, but the reason is pretty silly: we're only going to ever integrate f over D, which is an open disk, and hence the disk is simply connected anyways. The presence of 2πi, which you saw earlier in the form circle z -1 dz, is no accident. In fact, that's the central result we're going to use to prove the result.

Proof. There are several proofs out there, but I want to give the one that really draws out the power of Cauchy's theorem. Here's the picture we have: there's a point a sitting inside a circle γ, and we want to get our hands on the value f (a).

γ a

We're going to do a trick: construct a keyhole contour Γ δ,ε which has an outer circle γ, plus an inner circle γ ε , which is a circle centered at a with radius ε, running clockwise (so that γ ε runs counterclockwise). The "width" of the corridor is δ. See picture: 

The function f (z)

z-a manages to be holomorphic on all of Ω. Thus Cauchy's theorem applies and tells us that

0 = Γ δ,ε f (z) z -a dz.

As we let δ → 0, the two walls of the keyhole will cancel each other (because f is continuous, and the walls run in opposite directions). So taking the limit as δ → 0, we are left with just γ and γ ε , which (taking again orientation into account) gives

γ f (z) z -a dz = - γε f (z) z -a dz = γε f (z) z -a dz.

Thus we've managed to replace γ with a much smaller circle γ ε centered around a, and the rest is algebra.

To compute the last quantity, write

γε f (z) z -a dz = γε f (z) -f (a) z -a dz + f (a) • γε 1 z -a dz = γε f (z) -f (a) z -a dz + 2πif (a).

where we've used Theorem 31.3.1 Thus, all we have to do is show that

γε f (z) -f (a) z -a dz = 0.

For this we can basically use the weakest bound possible, the so-called M L lemma which I'll cite without proof: it says "bound the function everywhere by its maximum". 

f (a) = 1 2πi γ f (z) z -a dz = 1 2πi γ f (z)/z 1 -a z dz = 1 2πi γ f (z)/z • k≥0 a z k dz

You can prove (using the so-called Weierstrass M-test) that the summation order can be switched:

f (a) = 1 2πi k≥0 γ f (z) z • a z k dz = 1 2πi k≥0 γ a k • f (z) z k+1 dz = k≥0 1 2πi γ f (z) z k+1 dz a k . Letting c k = 1 2πi γ f (z)

z k+1 dz, and noting this is independent of a, this is

f (a) = k≥0 c k a k

and that's the miracle: holomorphic functions are given by a Taylor series! This is one of the biggest results in complex analysis. Moreover, if one is willing to believe that we can take the derivative k times, we obtain

c k = f (k) (0) k!

and this gives us

f (k) (0) = k! • c k .

Naturally, we can do this with any circle (not just one centered at zero). So let's state the full result below, with arbitrary center p. 

f (z) = c 0 + c 1 (z -p) + c 2 (z -p) 2 + . . .

where

c k = f k (p) k! = 1 2πi γ f (w -p) (w -p) k+1 dw

In particular,

f (k) (p) = k!c k = k! 2πi γ f (w -p) (w -p) k+1 dw.

Most importantly,

Over any disk, a holomorphic function is given exactly by a Taylor series.

This establishes a result we stated at the beginning of the chapter: that a function being complex differentiable once means it is not only infinitely differentiable, but in fact equal to its Taylor series. I should maybe emphasize a small subtlety of the result: the Taylor series centered at p is only valid in a disk centered at p which lies entirely in the domain U . If U = C this is no issue, since you can make the disk big enough to accommodate any point you want.

It's more subtle in the case that U is, for example, a square; you can't cover the entire square with a disk centered at some point without going outside the square. However, since U is open we can at any rate at least find some open neighborhood for which the Taylor series is correct -in stark contrast to the real case. Indeed, as you'll see in the problems, the existence of a Taylor series is incredibly powerful. §31.7 A few harder problems to think about These aren't olympiad problems, but I think they're especially nice! In the next complex analysis chapter we'll see some more nice applications.

The first few results are the most important. z . We are going to see that meromorphic functions behave like "almost-holomorphic" functions. Specifically, a meromorphic function A/B will be holomorphic at all points except the zeros of B (called poles). By the identity theorem, there cannot be too many zeros of B! So meromorphic functions can be thought of as "almost holomorphic" (like 1 z , which is holomorphic everywhere but the origin). We saw that

1 2πi γ 1 z dz = 1

for γ(t) = e it the unit circle. We will extend our results on contours to such situations. It turns out that, instead of just getting γ f (z) dz = 0 like we did in the holomorphic case, the contour integrals will actually be used to count the number of poles inside the loop γ. On the other hand, suppose p ∈ U has B(p) = 0: without loss of generality, p = 0 to ease notation. By using the Taylor series at p = 0 we can put

B(z) = c k z k + c k+1 z k+1 + . . .

with c k ̸ = 0 (certainly some coefficient is nonzero since B is not identically zero!). Then we can write 1

B(z) = 1 z k • 1 c k + c k+1 z + . . .

. It is meromorphic, because it is holomorphic everywhere except at the zeros of sin z.

But

f (z) = c -m (z -p) m + c -m+1 (z -p) m-1 + • • • + c -1 z -p + c 0 + c 1 (z -p) + . . .

At each of these points we can put a Laurent series: for example at z = 0 we have

z + 1 sin z = (z + 1) • 1 z -z 3 3! + z 5 5! -. . . = 1 z • z + 1 1 -z 2 3! -z 4 5! + z 6 7! -. . . = 1 z • (z + 1) k≥0 z 2 3! - z 4 5! + z 6 7! -. . . k .

If we expand out the horrible sum (which I won't do), then you get 1 z times a perfectly fine Taylor series, i.e. a Laurent series.

Abuse of Notation 32.2.4. We'll often say something like "consider the function f : C → C by z → 1 z ". Of course this isn't completely correct, because f doesn't have a value at z = 0. If I was going to be completely rigorous I would just set f (0) = 2015 or something and move on with life, but for all intents let's just think of it as "undefined at z = 0".

Why don't I just write g : C \ {0} → C? The reason I have to do this is that it's still important for f to remember it's "trying" to be holomorphic on C, even if isn't assigned a value at z = 0. As a function C \ {0} → C the function 1 z is actually holomorphic.

Remark 32.2.5 -I have shown that any function A(z)/B(z) has this characterization with poles, but an important result is that the converse is true too: if f : U \ S → C is holomorphic for some isolated set S, and moreover f admits a Laurent series at each point in S, then f can be written as a rational quotient of holomorphic functions. I won't prove this here, but it is good to be aware of.

Definition 32.2.6. Let p be a pole of a meromorphic function f , with Laurent series

f (z) = c -m (z -p) m + c -m+1 (z -p) m-1 + • • • + c -1 z -p + c 0 + c 1 (z -p) + . . . .

The integer m is called the order of the pole. A pole of order 1 is called a simple pole.

We also give the coefficient c -1 a name, the residue of f at p, which we write Res(f ; p).

The order of a pole tells you how "bad" the pole is. The order of a pole is the "opposite" concept of the multiplicity of a zero. If f has a pole at zero, then its Laurent series near z = 0 might look something like

f (z) = 1 z 5 + 8 z 3 - 2 z 2 + 4 z + 9 -3z + 8z 2 + . . .

and so f has a pole of order five. By analogy, if g has a zero at z = 0, it might look something like g(z) = 3z 3 + 2z 4 + 9z 5 + . . .

and so g has a zero of multiplicity three. These orders are additive: f (z)g(z) still has a pole of order 5 -3 = 2, but f (z)g(z) 2 is completely patched now, and in fact has a simple zero now (that is, a zero of degree 1). Metaphorically, poles can be thought of as "negative zeros".

We can now give many more examples. 

z z 2 = 1 z 2 + 1 z + 1 2 + z 6 + z 2 24 + z 3 120 + . . . .

Hence the pole z = 0 has order 2 and residue 1.

Example 32.2.9 (A rational meromorphic function)

Consider the function C → C given by 

z → z 4 + 1 z 2 -1 = z 2 + 1 + 2 (z -1)(z + 1) = z 2 + 1 + 1 z -1 • 1 1 + z-1 2 = 1 z -1 + 3 2 + 9 4 (z -1) + 7 8 (z -1) 2 -. . .

1 sin(2πz) = 1 z 1! -z 3 3! + z 5 5! -. . . = 1 z • 1 1 -z 2 3! -z 4 5! + . . . = 1 z k≥0 z 2 3! - z 4 5! + . . . k .

which is a Laurent series, though I have no clue what the coefficients are. You can at least see the residue; the constant term of that huge sum is 1, so the residue is 1. Also, the pole has order 1.

The Laurent series, if it exists, is unique (as you might have guessed), and by our result on holomorphic functions it is actually valid for any disk centered at p (minus the point p). The part c -1 z-p + • • • + c -m (z-p) m is called the principal part, and the rest of the series c 0 + c 1 (z -p) + . . . is called the analytic part.

§32.3 Winding numbers and the residue theorem

Recall that for a counterclockwise circle γ and a point p inside it, we had

γ (z -p) m dz = 0 m ̸ = -1 2πi m = -1

where m is an integer. One can extend this result to in fact show that γ (z -p) m dz = 0 for any loop γ, where m ̸ = -1. So we associate a special name for the nonzero value at m = -1.

Definition 32.3.1. For a point p ∈ C and a loop γ not passing through it, we define the winding number, denoted I(γ, p), by

I(γ, p) = 1 2πi γ 1 z -p dz

For example, by our previous results we see that if γ is a circle, we have I(circle, p) = 1 p inside the circle 0 p outside the circle.

If you've read the chapter on fundamental groups, then this is just the fundamental group associated to C \ {p}. In particular, the winding number is always an integer.

(Essentially, it uses the complex logarithm to track how the argument of the function changes. The details are more complicated, so we omit them here). In the simplest case the winding numbers are either 0 or 1.

Definition 32.3.2. We say a loop γ is regular if I(γ, p) = 1 for all points p in the interior of γ (for example, if γ is a counterclockwise circle).

With all these ingredients we get a stunning generalization of the Cauchy-Goursat theorem:

Theorem 32.3.3 (Cauchy's residue theorem)

Let f : Ω → C be meromorphic, where Ω is simply connected. Then for any loop γ not passing through any of its poles, we have

1 2πi γ f (z) dz = pole p I(γ, p) Res(f ; p).

In particular, if γ is regular then the contour integral is the sum of all the residues, in the form

1 2πi γ f (z) dz = pole p inside γ

Res(f ; p). The proof from here is not really too impressive -the "work" was already done in our statements about the winding number.

Proof. Let the poles with nonzero winding number be p 1 , . . . , p k (the others do not affect the sum). 1 Then we can write f in the form

f (z) = g(z) + k i=1 P i 1 z -p i where P i 1 z-p i is the principal part of the pole p i . (For example, if f (z) = z 3 -z+1 z(z+1) we would write f (z) = (z -1) + 1 z -1 1+z

.) The point of doing so is that the function g is holomorphic (we've removed all the "bad" parts), so γ g(z) dz = 0 by Cauchy-Goursat.

On the other hand, if 1 To show that there must be finitely many such poles: recall that all our contours γ : [a, b] → C are in fact bounded, so there is some big closed disk D which contains all of γ. The poles outside D thus have winding number zero. Now we cannot have infinitely many poles inside the disk D, for D is compact and the set of poles is a closed and isolated set! which gives the conclusion.

P i (x) = c 1 x + c 2 x 2 + • • • + c d x d then γ P i 1 z -p i dz = γ c 1 • 1 z -p i dz + γ c 2 • 1 z -p i 2 dz + . . . = c 1 • I(γ, p i ) + 0 + 0 + . . . = I(γ, p i ) Res(f ; p i ).

§32.4 Argument principle

One tricky application is as follows. Given a polynomial P (x) = (x-a 1 ) e 1 (x-a 2 ) e 2 . . . (xa n ) en , you might know that we have

P ′ (x) P (x) = e 1 x -a 1 + e 2 x -a 2 + • • • + e n x -a n .

The quantity P ′ /P is called the logarithmic derivative, as it is the derivative of log P . This trick allows us to convert zeros of P into poles of P ′ /P with order 1; moreover the residues of these poles are the multiplicities of the roots.

In an analogous fashion, we can obtain a similar result for any meromorphic function f . Proposition 32.4.1 (The logarithmic derivative)

Let f : U → C be a meromorphic function. Then the logarithmic derivative f ′ /f is meromorphic as a function from U to C; its zeros and poles are: (i) A pole at each zero of f whose residue is the multiplicity, and (ii) A pole at each pole of f whose residue is the negative of the pole's order.

Again, you can almost think of a pole as a zero of negative multiplicity. This spirit is exemplified below.

Proof. Dead easy with Taylor series. Let a be a zero/pole of f , and WLOG set a = 0 for convenience. We take the Taylor series at zero to get

f (z) = c k z k + c k+1 z k+1 + . . .

where k < 0 if 0 is a pole and k > 0 if 0 is a zero. Taking the derivative gives

f ′ (z) = kc k z k-1 + (k + 1)c k+1 z k + . . . . Now look at f ′ /f ; with some computation, it equals f ′ (z) f (z) = 1 z kc k + (k + 1)c k+1 z + . . . c k + c k+1 z + . . . .

So we get a simple pole at z = 0, with residue k.

Using this trick you can determine the number of zeros and poles inside a regular closed curve, using the so-called Argument Principle.

Theorem 32.4.2 (Argument principle)

Let γ be a regular curve. Suppose f : U → C is meromorphic inside and on γ, and none of its zeros or poles lie on γ. Then

1 2πi γ f ′ f dz = f rac12πi f •γ 1 z dz = Z -P

where Z is the number of zeros inside γ (counted with multiplicity) and P is the number of poles inside γ (again with multiplicity).

Proof. Immediate by applying Cauchy's residue theorem alongside the preceding proposition. In fact you can generalize to any curve γ via the winding number: the integral is

1 2πi γ f ′ f dz = zero z I(γ, z) - pole p I(γ, p)

where the sums are with multiplicity.

Thus the Argument Principle allows one to count zeros and poles inside any region of choice.

Computers can use this to get information on functions whose values can be computed but whose behavior as a whole is hard to understand. Suppose you have a holomorphic function f , and you want to understand where its zeros are. Then just start picking various circles γ. Even with machine rounding error, the integral will be close enough to the true integer value that we can decide how many zeros are in any given circle. Numerical evidence for the Riemann Hypothesis (concerning the zeros of the Riemann zeta function) can be obtained in this way. §32.5 Philosophy: why are holomorphic functions so nice?

All the fun we've had with holomorphic and meromorphic functions comes down to the fact that complex differentiability is such a strong requirement. It's a small miracle that C, which a priori looks only like R 2 , is in fact a field. Moreover, R 2 has the nice property that one can draw nontrivial loops (it's also true for real functions that a a f dx = 0, but this is not so interesting!), and this makes the theory much more interesting.

As another piece of intuition from Siu 

logarithms

In this chapter we'll make sense of a holomorphic square root and logarithm. The main results are Theorem 33.3.2, Theorem 33.4.2, Corollary 33.5.1, and Theorem 33.5.2. If you like, you can read just these four results, and skip the discussion of how they came to be.

Let z) for all z ∈ U . The main question we'll try to figure out is: when do these exist? In particular, what if f = id? §33.1 Motivation: square root of a complex number To start us off, can we define √ z for any complex number z? The first obvious problem that comes up is that for any z, there are two numbers w such that w 2 = z. How can we pick one to use? For our ordinary square root function, we had a notion of "positive", and so we simply took the positive root.

f : U → C be a holomorphic function. A holomorphic nth root of f is a function g : U → C such that f (z) = g(z) n for all z ∈ U . A logarithm of f is a function g : U → C such that f (z) = e g(

Let's expand on this: given z = r (cos θ + i sin θ) (here r ≥ 0) we should take the root to be w = √ r (cos α + i sin α) .

such that 2α ≡ θ (mod 2π); there are two choices for α (mod 2π), differing by π.

For complex numbers, we don't have an obvious way to pick α. Nonetheless, perhaps we can also get away with an arbitrary distinction: let's see what happens if we just choose the α with -1 2 π < α ≤ 1 2 π. Pictured below are some points (in red) and their images (in blue) under this "upperhalf" square root. The condition on α means we are forcing the blue points to lie on the right-half plane.

Im

Re 0

z 0 z 1 z 2 z 3 z 4 z 5 z 6 z 7 w 0 w 1 w 2 w 3 w 4 w 5 w 6 w 7

Here, w 2 i = z i for each i, and we are constraining the w i to lie in the right half of the complex plane. We see there is an obvious issue: there is a big discontinuity near the points w 5 and w 7 ! The nearby point w 6 has been mapped very far away. This discontinuity occurs since the points on the negative real axis are at the "boundary". For example, given -4, we send it to -2i, but we have hit the boundary: in our interval -1 2 π ≤ α < 1 2 π, we are at the very left edge. The negative real axis that we must not touch is what we will later call a branch cut, but for now I call it a ray of death. It is a warning to the red points: if you cross this line, you will die! However, if we move the red circle just a little upwards (so that it misses the negative real axis) this issue is avoided entirely, and we get what seems to be a "nice" square root.

Re 0

z 0 z 1 z 2 z 3 z 4 z 5 z 6 z 7 w 0 w 1 w 2 w 3 w 4 w 5 w 6 w 7

In fact, the ray of death is fairly arbitrary: it is the set of "boundary issues" that arose when we picked -1 2 π < α ≤ 1 2 π. Suppose we instead insisted on the interval 0 ≤ α < π; then the ray of death would be the positive real axis instead. The earlier circle we had now works just fine.

Re 0

z 0 z 1 z 2 z 3 z 4 z 5 z 6 z 7 w 0 w 1 w 2 w 3 w 4 w 5 w 6 w 7

What we see is that picking a particular α-interval leads to a different set of edge cases, and hence a different ray of death. The only thing these rays have in common is their starting point of zero. In other words, given a red circle and a restriction of α, I can make a nice "square rooted" blue circle as long as the ray of death misses it.

So, what exactly is going on? §33.

2 Square roots of holomorphic functions

To get a picture of what's happening, we would like to consider a more general problem: let f : U → C be holomorphic. Then we want to decide whether there is a holomorphic g : U → C such that

f (z) = g(z) 2 .

Our previous discussion with f = id tells us we cannot hope to achieve this for U = C; there is a "half-ray" which causes problems. However, there are certainly functions f : C → C such that a g exists. As a simplest example, f (z) = z 2 should definitely have a square root! Now let's see if we can fudge together a square root. Earlier, what we did was try to specify a rule to force one of the two choices at each point. This is unnecessarily strict. Perhaps we can do something like: start at a point in z 0 ∈ U , pick a square root w 0 of f (z 0 ), and then try to "fudge" from there the square roots of the other points. What do I mean by fudge? Well, suppose z 1 is a point very close to z 0 , and we want to pick a square root w 1 of f (z 1 ). While there are two choices, we also would expect w 0 to be close to w 1 . Unless we are highly unlucky, this should tell us which choice of w 1 to pick. (Stupid concrete example: if I have taken the square root -4.12i of -17 and then ask you to continue this square root to -16, which sign should you pick for ±4i?)

There are two possible ways we could get unlucky in the scheme above: first, if w 0 = 0, then we're sunk. But even if we avoid that, we have to worry that if we run a full loop in the complex plane, we might end up in a different place from where we started. For concreteness, consider the following situation, again with f = id:

Im Re 0 z 0 z 1 z 2 z 3 z 4 z 5 z 6 z 7 w 0 w 1 w 2 w 3 w 4 w 5 w 6 w 7

We started at the point z 0 , with one of its square roots as w 0 . We then wound a full red circle around the origin, only to find that at the end of it, the blue arc is at a different place where it started!

The interval construction from earlier doesn't work either: no matter how we pick the interval for α, any ray of death must hit our red circle. The problem somehow lies with the fact that we have enclosed the very special point 0.

Nevertheless, we know that if we take f (z) = z 2 , then we don't run into any problems with our "make it up as you go" procedure. So, what exactly is going on? §33.3 Covering projections By now, if you have read the part on algebraic topology, this should all seem quite familiar. The "fudging" procedure exactly describes the idea of a lifting.

More precisely, recall that there is a covering projection 

(-) 2 : C \ {0} → C \ {0}. Let V = {z ∈ U | f (z) ̸ = 0}. For z ∈ U \ V ,

E = C \ {0} V B = C \ {0}. p=• 2 g f

Our map p can be described as "winding around twice". Our Theorem 60.2.5 now tells us that this lifting exists if and only if

f img * (π 1 (V )) ⊆ p img * (π 1 (E)

) is a subset of the image of π 1 (E) by p. Since B and E are both punctured planes, we can identify them with S 1 .

Question 33.3.1. Show that the image under p is exactly 2Z once we identify π 1 (B) = Z.

That means that for any loop γ in V , we need f • γ to have an even winding number around 0 ∈ B. This amounts to

1 2π γ f ′ f dz ∈ 2Z

since f has no poles.

Replacing 2 with n and carrying over the discussion gives the first main result.

Theorem 33.3.2 (Existence of holomorphic nth roots)

Let f : U → C be holomorphic. Then f has a holomorphic nth root if and only if

1 2πi γ f ′ f dz ∈ nZ

for every contour γ in U .

§33.4 Complex logarithms

The multivalued nature of the complex logarithm comes from the fact that exp(z + 2πi) = exp(z).

So if e w = z, then any complex number w + 2πik is also a solution.

We can handle this in the same way as before: it amounts to a lifting of the following diagram.

E = C U B = C \ {0} p=exp g f

There is no longer a need to work with a separate V since: Finally, let's return to the question of f = id from the very beginning. What's the best domain U such that √ -: U → C is well-defined? Clearly U = C cannot be made to work, but we can do almost as well.

For note that the only zero of f = id is at the origin. Thus if we want to make a logarithm exist, all we have to do is make an incision in the complex plane that renders it impossible to make a loop around the origin. The usual choice is to delete negative half of the real axis, our very first ray of death; we call this a branch cut, with branch point at 0 ∈ C (the point which we cannot circle around). This gives Theorem 33.

5.2 (Branch cut functions)

There exist holomorphic functions

log : C \ (-∞, 0] → C n √ -: C \ (-∞, 0] → C

satisfying the obvious properties.

There are many possible choices of such functions (n choices for the nth root and infinitely many for log); a choice of such a function is called a branch. So this is what is meant by a "branch" of a logarithm. The principal branch is the "canonical" branch, analogous to the way we arbitrarily pick the positive branch to define √ -: R ≥0 → R ≥0 . For log, we take the w such that e w = z and the imaginary part of w lies in (-π, π] (since we can shift by integer multiples of 2πi). Often, authors will write Log z to emphasize this choice. 34 Bonus: Topological Abel-Ruffini Theorem We've already shown the Fundamental Theorem of Algebra. Now, with our earlier intuition on holomorphic nth roots, we can now show that there is no general formula for the roots of a quintic polynomial.

§34.1 The Game Plan

Firstly, what do we even mean by "formula" here? Definition 34.1.1. A quintic formula would be a formula taking in the coefficients (a 0 , . . . , a 5 ) of a degree 5 polynomial P , using the operations +, -, ×, ÷, n √ finitely many times, that maps to the five roots (z 1 , • • • , z 5 ) of P . Now, any proposed quintic formula F receives the same coefficients when the roots are the same, and thus gives the same output. This is fine at first glance, but swapping two roots continuously might pose more issues. F must create and preserve some order of the roots under these permutations. √ operator for motivation. Suppose I told you that some rational function R always finds a root of a quintic polynomial

P (z) = (z -z 1 )(z -z 2 )(z -z 3 )(z - z 4 )(z -z 5 ).

For simplicity, let all the roots be distinct.

Suppose that initially R outputs z 1 . Consider what happens we smoothly swap the roots z 1 and z 2 along two non-intersecting paths that doesn't go through other roots.

Im

Re 0

z 1 z 2 z 3 z 4 z 5

Since R is continuous, it must be tracking the same root. However, once we finish swapping z 1 and z 2 , the coefficients of P are the same as they were initially. But this means that R has been tricked into changing the root it outputs, contradiction!

The bigger picture here is that we were able to find an operation that fixes R while changing the order of the roots in S 5 .

§34.3 Step 2: Nested Roots

Once we add n √ back to the picture, this idea no longer works right out of the box.

Example 34.3.1 (Quadratic Formula)

If you've done any competition math, you know that for a polynomial P (z) = az 2 + bz + c = (z -z 1 )(z -z 2 ), it follows that the two branches of

-b ± √ b 2 -4ac 2a

give z 1 and z 2 . So why can't swapping z 1 and z 2 yield a contradiction here? It's because while all the coefficients end up in the starting position, the liftings of how √ b 2 -4ac travels may not.

Exercise 34.3.2. Consider the polynomial z 2 -1. Then smoothly swap the roots to get the intermediary polynomials of (z -e it )(z + e it ). See that the two roots given by the quadratic formula also swap position.

Let's now consider the next simplest case of the nth root of a rational function n √ R, and try to fix it with a nontrivial permutation of the roots.

Swapping the roots z 1 and z 2 , we keep R the same, but R's path α around the origin may have accumulated some change in phase 2πa. If we were to unswap z 1 and z 2 in the same manner, we'd undo the change in phase, but we'd also be back to doing nothing.

However, while changes in phase are abelian, permutations are not. Let's consider another operation of swapping the roots z 2 and z 3 . Taking a commutator of the two operations, we keep all the phases the same, but end up with a permutation (12)(23)(12) -1 (23) -1 .

If we mark the second operation's path with β, this corresponds to αβα -1 β -1 . We now have better tools: We have permutations in S 5 that fix the nth roots of rational functions, and their compositions under +, -, ×, ÷.

Im

How do we handle the nested radicals now?

Example 34.3.4 (Cubic Formula)

The cubic formula contains a nasty term

3 2b 3 -9abc + 27a 2 d + (2b 3 -9abc + 27a d ) 2 -4(b 3 -3ac) 3 2 .

Here, we've taken multiple roots.

Definition 34.3.5. Define the degree of a nested radical as the maximum number of times radicals can be found in other radicals.

Let's now consider nested radicals of degree 2, such as say

3 √ ab + c - √ d.

We know that we have nontrivial commutators σ and ρ that fix the interior of the cube root, but once again the phase may not be preserved under each operation individually. Once again, we can again consider the commutators of these commutators, say σρσ -1 ρ -1 which by the same logic fixes the issues with phase.

There's no reason, we can't consider the commutators of commutators of commutators to fix radicals of degree 3 and so on. It thus just remains that we always can keep getting nontrivial commutators.

§34.4 Step 3: Normal Groups

We've reduced this to a group theory problem. Given a chain of commutators

S 5 = G ⊇ G (1) ⊇ G (2) ⊇ . . .

where each group is the commutator subgroup of the next, we want to show that G (n) never becomes trivial. This chain is called the derived series. So all that remains is showing that S 5 is not solvable. This is a calculation that isn't relevant to the topology ideas in this chapter, so we defer it to Problem 34A. §34.5 Summary While this is indeed a valid proof, it has some pros and cons. As a con, we haven't shown that any polynomial such as z 5 -z -1 has a root that can't be expressed using nested nth roots. We've only that we don't have a formula for all degree 5 polynomials.

As a pro, this argument makes it easy to add even more functions such as exp, sin, and cos to the mix and show even then that no such formula exists. It also allows you to broadly understand what people mean when they compare this theorem to a fact that A 5 is not solvable. §34.6 A few harder problems to think about Problem 34A. Show that A 5 is not solvable. 

36

Constructing the Borel and Lebesgue measure 375 36.1 Pre-measures . 

37

Lebesgue integration 385 37.1 The definition . 

38

Swapping order with Lebesgue integrals 389 38.1 Motivating limit interchange . 

39

Bonus: A hint of Pontryagin duality 393 39.1 LCA groups . 

39.3

The orthonormal basis in the compact case . . . 39.4 The Fourier transform of the non-compact case . 

Here is an outline of where we are going next. Our goal over the next few chapters is to develop the machinery to state (and in some cases prove) the law of large numbers and the central limit theorem. For these purposes, the scant amount of work we did in Calculus 101 is going to be awfully insufficient: integration over R (or even R n ) is just not going to cut it. This chapter will develop the theory of "measure spaces", which you can think of as "spaces equipped with a notion of size". We will then be able to integrate over these with the so-called Lebesgue integral (which in some senses is almost strictly better than the Riemann one).

§35.1 Letter connotations

There are a lot of "types" of objects moving forward, so here are the letter connotations we'll use throughout the next several chapters. This makes it easier to tell what the "type" of each object is just by which letter is used.

• Measure spaces denoted by Ω, their elements denoted by ω.

• Algebras and σ-algebras denoted by script A , B, . . . . Sets in them denoted by early capital Roman A, B, C, D, E, . . . .

• Measures (i.e. functions assigning sets to reals) denoted usually by µ or ρ.

• Random variables (functions sending worlds to reals) denoted usually by late capital Roman X, Y , Z, . . . .

• Functions from R → R by Roman letters like f and g for pdf's and F and G for cdf's.

• Real numbers denoted by lower Roman letters like x, y, z.

§35.2 Motivating measure spaces via random variables

To motivate why we want to construct measure spaces, I want to talk about a (real) random variable, which you might think of as

• the result of a coin flip,

• the high temperature in Boston on Saturday,

• the possibility of rain on your 18.725 date next weekend.

Why does this need a long theory to develop well? For a simple coin flip one intuitively just thinks "50% heads, 50% tails" and is done with it. The situation is a little trickier with temperature since it is continuous rather than discrete, but if all you care about is that one temperature, calculus seems like it might be enough to deal with this.

But it gets more slippery once the variables start to "talk to" each other: the high temperature tells you a little bit about whether it will rain, because e.g. if the temperature is very high it's quite likely to be sunny. Suddenly we find ourselves wishing we could talk about conditional probability, but this is a whole can of worms -the relations between these sorts of things can get very complicated very quickly.

The big idea to getting a formalism for this is that:

Our measure spaces Ω will be thought of as a space of entire worlds, with each ω ∈ Ω representing a world. Random variables are functions from worlds to R.

This way, the space of "worlds" takes care of all the messy interdependence. Then, we can assign "measures" to sets of worlds: for example, to be a fair coin means that if you are only interested in that one coin flip, the "fraction" of worlds in which that coin showed heads should be 1 2 . This is in some ways backwards from what you were told in high-school: officially, we start with the space of worlds, rather than starting with the probabilities.

It will soon be clear that there is no way we can assign a well-defined measure to every single one of the 2 Ω subsets. Fortunately, in practice, we won't need to, and the notion of a σ-algebra will capture the idea of "enough measur-able sets for us to get by".

Remark 35.2.1 (Random seeds) -Another analogy if you do some programming: each ω ∈ Ω is a random seed, and everything is determined from there. §35.3 Motivating measure spaces geometrically So, we have a set Ω of possible points (which in the context of the previous discussion can be thought of as the set of worlds), and we want to assign a measure (think volume) to subsets of points in Ω. We will now describe some of the obstacles that we will face, in order to motivate how measure spaces are defined (as the previous section only motivated why we want such things).

If you try to do this naïvely, you basically immediately run into set-theoretic issues. A good example to think about why this might happen is if Ω = R 2 with the measure corresponding to area. You can define the area of a triangle as in high school, and you can then try and define the area of a circle, maybe by approximating it with polygons. But what area would you assign to the subset Q 2 , for example? (It turns out "zero" is actually a working answer.) Or, a unit disk is composed of infinitely many points; each of the points better have measure zero, but why does their union have measure π then? Blah blah blah.

We'll say more about this later, but you might have already heard of the Banach-Tarski paradox which essentially shows there is no good way that you can assign a measure to every single subset of R 3 and still satisfy basic sanity checks. There are just too many possible subsets of Euclidean space.

However, the good news is that most of these sets are not ones that we will ever care about, and it's enough to define measures for certain "sufficiently nice sets". The adjective we will use is measurable, and it will turn out that this will be way, way more than good enough for any practical purposes.

We will generally use A, B, . . . for measurable sets and denote the entire family of measurable sets by curly A .

§35.4 σ-algebras and measurable spaces

Here's the machine code.

Definition 35.4.1.

A measurable space consists of a space Ω of points, and a σalgebra A of subsets of Ω (the "measurable sets" of Ω). The set A is required to satisfy the following axioms:

• A contains ∅ and Ω.

• A should be closed under complements and countable unions/intersections. (Hint on nomenclature: σ usually indicates some sort of "countably finite" condition.)

(Complaint: this terminology is phonetically confusing, because it can be confused with "measure space" later. The way to think about is that "measurable spaces have a σ-algebra, so we could try to put a measure on it, but we haven't, yet.") Though this definition is how we actually think about it in a few select cases, for the most part, and we will usually instantiate A in practice in a different way: Definition 35.4.2. Let Ω be a set, and consider some family of subsets F of Ω. Then the σ-algebra generated by F is the smallest σ-algebra A which contains F .

As is commonplace in math, when we see "generated", this means we sort of let the definition "take care of itself". So, if Ω = R, maybe I want A to contain all open sets. Well, then the definition means it should contain all complements too, so it contains all the closed sets. Then it has to contain all the half-open intervals too, and then. . (a) If Ω is any set, then the power set A = 2 Ω is obviously a σ-algebra. This will be used if Ω is countably finite, but it won't be very helpful if Ω is huge.

(b) If Ω is an uncountable set, then we can declare A to be all subsets of Ω which are either countable, or which have countable complement. (You should check this satisfies the definitions.) This is a very "coarse" algebra. 

µ n A n = n µ(A n ).

The triple (Ω, A , µ) is called a measure space. It's called a probability space if µ(Ω) = 1.

Exercise 35.5.2 (Weaker equivalent definitions). I chose to give axioms for A and µ that capture how people think of them in practice, which means there is some redundancy: for example, being closed under complements and unions is enough to get intersections, by de Morgan's law. Here are more minimal definitions, which are useful if you are trying to prove something satisfies them to reduce the amount of work you have to do:

(a) The axioms on A can be weakened to (i) ∅ ∈ A and (ii) A is closed under complements and countable unions.

(b) The axioms on µ can be weakened to (i)

µ(∅) = 0, (ii) µ(A ⊔ B) = µ(A) + µ(B), and (iii) for A 1 ⊇ A 2 ⊇ • • • , we have µ ( n A n ) = lim n µ(A n ).

Remark 35.5.3 -Here are some immediate remarks on these definitions.

• If A ⊆ B are measurable, then µ(A) ≤ µ(B) since µ(B) = µ(A) + µ(B -A).

• In particular, in a probability space all measures are in [0, 1]. On the other hand, for general measure spaces we'll allow +∞ as a possible measure (hence the choice of [0, +∞] as codomain for µ).

• We want to allow at least countable unions / additivity because with finite unions it's too hard to make progress: it's too hard to estimate the area of a circle without being able to talk about limits of countably infinitely many triangles.

We don't want to allow uncountable unions and additivity, because uncountable sums basically never work out. In particular, there is a nice elementary exercise as follows:

Exercise 35.5.4 (Tricky). Let S be an uncountable set of positive real numbers. Show that some finite subset T ⊆ S has sum greater than 10 2019 . Colloquially, "uncountably many positive reals cannot have finite sum".

So countable sums are as far as we'll let the infinite sums go. This is the reason why we considered σ-algebras in the first place. This is useful if Ω is countably infinite, and optimal if Ω is finite (and nonempty).

In the latter case, we will often normalize by µ(A) = |A| |Ω| so that Ω becomes a probability space.

(b) Suppose Ω was uncountable and we took A to be the countable sets and their complements. Then Let Ω denote the unit circle in R 2 and A = 2 Ω . We will show that any measure µ on Ω with µ(Ω) = 1 will have undesirable properties.

µ(A) = 0 A is countable 1 Ω \ A is countable

Let ∼ denote an equivalence relation on Ω defined as follows: two points are equivalent if they differ by a rotation around the origin by a rational multiple of π. We may pick a representative from each equivalence class, letting X denote the set of representatives. Then Ω = q∈Q 0≤q<2 (X rotated by qπ radians) .

Since we've only rotated X, each of the rotations should have the same measure m. But µ(Ω) = 1, and there is no value we can assign that measure: if m = 0 we get µ(Ω) = 0 and m > 0 we get µ(Ω) = ∞.

Remark 35.6.2 (Choice) -Experts may recognize that picking a representative (i.e. creating set X) technically requires the Axiom of Choice. That is why, when people talk about Banach-Tarski issues, the Axiom of Choice almost always gets honorable mention as well.

Stay tuned to actually see a construction for B(R n ) in the next chapter.

§35.7 Measurable functions

In the past, when we had topological spaces, we considered continuous functions. The analog here is: Definition 35.7.1. Let (X, A ) and (Y, B) be measurable spaces (or measure spaces). A function f : X → Y is measurable if for any measurable set S ⊆ Y (i.e. S ∈ B) we have f pre (S) is measurable (i.e. f pre (S) ∈ A ).

In practice, most functions you encounter will be continuous anyways, and in that case we are fine. In later chapters we will begin seeing the phrase "almost everywhere" and "almost surely" start to come up, and it seems prudent to take the time to talk about it now. For example, if we say "f = g almost everywhere" for some functions f and g defined on a measure space Ω, then we mean that f (ω) = g(ω) for all ω ∈ Ω other than a measure-zero set.

There, that's the definition. The main thing to now update your instincts on is that

In measure theory, we basically only care about things up to almosteverywhere.

Here are some examples:

• If f = g almost everywhere, then measure theory will basically not tell these functions apart. For example, Ω f dω = Ω g dω will hold for two functions agreeing almost everywhere.

• As another example, if we prove "there exists a unique function f such that so-and-so", the uniqueness is usually going to be up to measure-zero sets.

You can think of this sort of like group isomorphism, where two groups are considered "basically the same" when they are isomorphic, except this one might take a little while to get used to. • Countable additivity: if A 1 , A 2 , . . . are disjoint sets in A 0 and moreover the disjoint union A i is contained in A 0 (not guaranteed by algebra axioms!), then

µ 0 n A n = n µ 0 (A n ).

Example 36. 1.4 (The pre-measure on R n )

Let Ω = R 2 . Then, let L 0 be the algebra generated by rectangles

[a 1 , a 2 ] × [b 1 , b 2 ]. We then let µ 0 ([a 1 , a 2 ] × [b 1 , b 2 ]) = (a 2 -a 1 )(b 2 -b 1 )

the area of the rectangle. As elements of L 0 are simply finite unions of rectangles and their complements (picture drawn earlier), it's not difficult to extend this to a pre-measure λ 0 which behaves as you expect -although we won't do this.

Since we are sweeping something under the rug that turns out to be conceptually important, I'll go ahead and blue-box it.

Proposition 36. 1.5 (Geometry sanity check that we won't prove)

For Ω = R n and L 0 the algebra generated by rectangular prisms, one can define a pre-measure λ 0 on L 0 .

From this point forwards, we will basically do almost no geometry 1 whatsoever in defining the measure B(R n ), and only use set theory to extend our measure. So, Proposition 36. 1.5 is the only sentry which checks to make sure that our "initial definition" is sane.

To put the point another way, suppose an insane scientist 2 tried to define a notion of area in which every rectangle had area 1. Intuitively, this shouldn't be possible: every rectangle can be dissected into two halves and we ought to have 1 + 1 ̸ = 1. However, the only thing that would stop them is that they couldn't extend their pre-measure on the algebra L 0 . If they somehow got past that barrier and got a pre-measure, nothing in the rest of the section would prevent them from getting an entire bona fide measure with this property. Thus, in our construction of the Lebesgue measure, most of the geometric work is captured in the (omitted) proof of Proposition 36. 1.5. 

§36.2 Outer measures

Prototypical example for this section: Keep taking Ω = R 2 ; see the picture to follow.

The other way to weaken a measure is to relax the countable additivity, and this yields the following: Definition 36.2.1. An outer measure µ * on a set Ω is a function µ * : 2 Ω → [0, +∞] satisfying the following axioms:

• µ * (∅) = 0;

1 White lie. Technically, we will use one more fact: that open sets of R n can be covered by countably infinitely many rectangles, as in Exercise 36.5.1. This step doesn't involve any area assignments, though. 2 Because "mad scientists" are overrated.

• if E ⊆ F and E, F ∈ 2 Ω then µ * (E) ≤ µ * (F );

• for any subsets E 1 , E 2 , . . . of Ω we have

µ * n E n ≤ n µ * (E n ).

(I don't really like the word "outer measure", since I think it is a bit of a misnomer: I would rather call it "fake measure", since it's not a measure either.)

The reason for the name "outer measure" is that you almost always obtain outer measures by approximating them from "outside" sets. Officially, the result is often stated as follows (as Problem 36A † ).

For a set Ω, let E be any subset of 2 Ω and let ρ : E → [0, +∞] be any function. Then

µ * (E) = inf ∞ n=1 ρ(E n ) | E n ∈ E, E ⊆ ∞ n=1 E n

is an outer measure.

However, I think the above theorem is basically always wrong to use in practice, because it is way too general. As I warned with the insane scientist, we really do want some sort of sanity conditions on ρ: otherwise, if we apply the above result as stated, there is no guarantee that µ * will be compatible with ρ in any way.

So, I think it is really better to apply the theorem to pre-measures µ 0 for which one does have some sort of guarantee that the resulting µ * is compatible with µ 0 . In practice, this is always how we will want to construct our outer measures.

Theorem 36.2.2 (Constructing outer measures from pre-measures) Let µ 0 be a pre-measure on an algebra A 0 on a set Ω.

(a) The map µ * : 2 Ω → [0, +∞] defined by

µ * (E) = inf ∞ n=1 µ 0 (A n ) | A n ∈ A 0 , E ⊆ ∞ n=1

A n is an outer measure.

(b) Moreover, this measure agrees with µ 0 on sets in A 0 .

Intuitively, what is going on is that µ * (A) is the infimum of coverings of A by countable unions of elements in A 0 . Part (b) is the first half of the compatibility condition I promised; the other half appears later as Proposition 36.3.2.

Proof of Theorem 36.2.2. As alluded to already, part (a) is a special case of Problem 36A † (and proving it in this generality is actually easier, because you won't be distracted by unnecessary properties).

We now check (b), that µ * (A) = µ 0 (A) for A ∈ A 0 . One bound is quick:

Question 36.2.3. Show that µ * (A) ≤ µ 0 (A).

For the reverse, suppose that A ⊆ n A n . Then, define the sets

B 1 = A ∩ A 1 B 2 = (A ∩ A 2 ) \ B 1 B 3 = (A ∩ A 3 ) \ (B 1 ∪ B 2 ) . . .

and so on. Then the B n are disjoint elements of A 0 with B n ⊂ A n , and we have rigged the definition so that n B n = A. Thus by definition of pre-measure,

µ 0 (A) = n µ 0 (B n ) ≤ n µ 0 (A n ) as desired.

Example 36.

2.4

Let Ω = R 2 and λ 0 the pre-measure from before. Then λ * (A) is, intuitively, the infimum of coverings of the set A by rectangles. Here is a picture you might use to imagine the situation with A being the unit disk.

circles covered by rectangles

Missing figure §36.3 Carathéodory extension for outer measures

We will now take any outer measure and turn it into a proper measure. To do this, we first need to specify the σ-algebra on which we will define the measure.

Definition 36.3.1. Let µ * be an outer measure. We say a set A is Carathéodory measurable with respect to µ * , or just µ * -measurable, if the following condition holds: for any set

E ∈ 2 Ω , µ * (E) = µ * (E ∩ A) + µ * (E \ A).

This definition is hard to motivate, but turns out to be the right one. One way to motivate is this: it turns out that in R n , it will be equivalent to a reasonable geometric condition (which I will state in Proposition 36. 4 .3), but since that geometric definition requires information about R n itself, this is the "right" generalization for general measure spaces.

Since our goal was to extend our A 0 , we had better make sure this definition lets us measure the initial sets that we started with! Proposition 36.3.2 (Carathéodory measurability is compatible with the initial A 0 ) Suppose µ * was obtained from a pre-measure µ 0 on an algebra A 0 , as in Theorem 36.2.2. Then every set in A 0 is µ * -measurable. This is the second half of the compatibility condition that we get if we make sure our initial µ 0 at least satisfies the pre-measure axioms. (The first half was (b) of Theorem 36.2.2.)

Proof. Let A ∈ A 0 and E ∈ 2 Ω ; we wish to prove µ * (E) = µ * (E ∩ A) + µ * (E \ A).

The definition of outer measure already requires µ * (E) ≤ µ * (E ∩ A) + µ * (E \ A) and so it's enough to prove the reverse inequality.

By definition of infimum, for any ε > 0, there is a covering

E ⊂ n A n with µ * (E)+ε ≥ n µ 0 (A n ). But n µ 0 (A n ) = n (µ 0 (A n ∩ A) + µ 0 (A n \ A)) ≥ µ * (E ∩ A) + µ * (E \ A)

with the first equality being the definition of pre-measure on A 0 , the second just being by definition of µ * (since

A n ∩ A certainly covers E ∩ A, for example). Thus µ * (E) + ε ≥ µ * (E ∩ A) + µ * (E \ A).

Since the inequality holds for any ε > 0, we're done.

To add extra icing onto the cake, here is one more niceness condition which our constructed measure will happen to satisfy.

Definition 36.3.3. A null set of a measure space (Ω, A , µ) is a set A ∈ A with µ(A) = 0. A measure space (Ω, A , µ) is complete if whenever A is a null set, then all subsets of A are in A as well (and hence null sets). This is a nice property to have, for obvious reasons. Visually, if I have a bunch of dust which I already assigned weight zero, and I blow away some of the dust, then the remainder should still have an assigned weight -zero. The extension theorem will give us σ-algebras with this property.

Theorem 36.3.4 (Carathéodory extension theorem for outer measures)

If µ * is an outer measure, and A cm is the set of µ * -measurable sets with respect to µ * , then A cm is a σ-algebra on Ω, and the restriction µ cm of µ * to A cm gives a complete measure space.

(Phonetic remark: you can think of the superscript cm as standing for either "Carathéodory measurable" or "complete". Both are helpful for remembering what this represents. This notation is not standard but the pun was too good to resist.)

Thus, if we compose Theorem 36.2.2 with Theorem 36.3.4, we find that every premeasure µ 0 on an algebra A 0 naturally gives a σ-algebra A cm with a complete measure µ cm , and our two compatibility results (namely (b) of Theorem 36.2.2, together with Proposition 36.3.2) means that A cm ⊃ A 0 and µ cm agrees with µ.

Here is a table showing the process, where going down each row of the table corresponds to restriction process.

Construct order Notes

2 Ω µ *

Step 2 µ * is outer measure obtained from µ 0

A cm µ cm

Step 3 A cm defined as µ * -measurable sets, (A cm , µ cm ) is complete.

A 0 µ 0 Step 1 µ 0 is a pre-measure §36.

4 Defining the Lebesgue measure

This lets us finally define the Lebesgue measure on R n . We wrap everything together at once now.

Definition 36.4.1. We create a measure on R n by the following procedure.

• Start with the algebra L 0 generated by rectangular prisms, and define a pre-measure λ 0 on this L 0 (this was glossed over in the example).

• By Theorem 36.2.2, this gives the Lebesgue outer measure λ * on 2 R n , which is compatible on all the rectangular prisms.

• By Carathéodory (Theorem 36. 3.4 ), this restricts to a complete measure λ on the σ-algebra L(R n ) of λ * -measurable sets (which as promised contains all rectangular prisms). 3The resulting complete measure, denoted λ, is called the Lebesgue measure.

The algebra L(R n ) we obtained will be called the Lebesgue σ-algebra; sets in it are said to be Lebesgue measurable.

Here is the same table from before, with the values filled in for the special case Ω = R n , which gives us the Lebesgue algebra.

Construct order Notes

2 R n λ *

Step 2 λ * is Lebesgue outer measure

L(R n ) λ

Step 3 Lebesgue σ-algebra (complete)

L 0 λ 0

Step 1 Define pre-measure on rectangles

Of course, now that we've gotten all the way here, if we actually want to compute any measures, we can mostly gleefully forget about how we actually constructed the measure and just use the properties. The hard part was to showing that there is a way to assign measures consistently; actually figuring out what that measure's value is given that it exists is often much easier. Here is an example.

Example 36.4.2 (The Cantor set has measure zero)

The standard middle-thirds Cantor set is the subset [0, 1] obtained as follows: we first delete the open interval (1/3, 2/3). This leaves two intervals [0, 1/3] and [2/3, 1] from which we delete the middle thirds again from both, i.e. deleting (1/9, 2/9) and (7/9, 8/9). We repeat this procedure indefinitely and let C denote the result. An illustration is shown below.

Image from [1207]

It is a classic fact that C is uncountable (it consists of ternary expansions omitting the digit 1). But it is measurable (it is an intersection of closed sets!) and we contend it has measure zero. Indeed, at the nth step, the result has measure (2/3) n leftover.

So µ(C) ≤ (2/3) n for every n, forcing µ(C) = 0. This is fantastic, but there is one elephant in the room: how are the Lebesgue σ-algebra and the Borel σ-algebra related? To answer this question briefly, I will state two results (but another answer is given in the next section). The first is a geometric interpretation of the strange Carathéodory measurable hypothesis.

Proposition 36.4.3 (A geometric interpretation of Lebesgue measurability)

A set A ⊆ R n is Lebesgue measurable if and only if for every ε > 0, there is an open set U ⊃ A such that λ * (U \ A) < ε

where λ * is the Lebesgue outer measure.

I want to say that this was Lebesgue's original formulation of "measurable", but I'm not sure about that. However, the containment is proper: there are more Lebesgue measurable sets than Borel ones. Indeed, it can actually be proven using transfinite induction (though we won't) that |B(R)| = |R|. Using this, one obtains: Exercise 36.4.5. Show the Borel σ-algebra is not complete. (Hint: consider the Cantor set. You won't be able to write down an example of a non-measurable set, but you can use cardinality arguments.) Thus the Lebesgue σ-algebra strictly contains the Borel one.

Nonetheless, there is a great way to describe the Lebesgue σ-algebra, using the idea of completeness.

Definition 36.4.6. Let (Ω, A , µ) be a measure space. The completion (Ω, A , µ) is defined as follows: we let

A = {A ∪ N | A ∈ A , N subset of null set} . and µ(A ∪ N ) = µ(A).

One can check this is well-defined, and in fact µ is the unique extension of µ from A to A .

This looks more complicated than it is. Intuitively, all we are doing is "completing" the measure by telling µ to regard any subset of a null set as having measure zero, too.

Then, the saving grace: Theorem 36.4.7 (Lebesgue is completion of Borel) For R n , the Lebesgue measure is the completion of the Borel measure.

Proof. This actually follows from results in the next section, namely Exercise 36.5.1 and part (c) of Carathéodory for pre-measures (Theorem 36.5.5).

Construct order Notes

2 R n λ *

Step 2 λ * is Lebesgue outer measure

L(R n ) λ

Step 3 Lebesgue σ-algebra, completion of Borel one

B(R n ) µ

Step 4 Borel σ-algebra, generated by L 0 L 0 λ 0

Step 1 Define pre-measure on rectangles

Going down one row of the table corresponds to restriction, while each of µ 0 → µ → µ cm is a unique extension when µ 0 is σ-finite.

Proof of Theorem 36. 5 . 5 . For (a): this is just Theorem 36.2.2 and Theorem 36. This integral may be +∞ (even if f is finite). As the details of the construction won't matter for us later on, we will state the relevant definitions, skip all the proofs, and also state all the properties that we actually care about. Consequently, this chapter will be quite short.

µ * (E) = inf ∞ n=1 ρ(E n ) | E n ∈ E, E ⊆ ∞ n=1 E n is an outer measure.

§37.1 The definition

The construction is done in four steps.

Definition 37.1.1. If A is a measurable set of Ω, then the indicator function 1 A : Ω → R is defined by

1 A (ω) = 1 ω ∈ A 0 ω / ∈ A.

Step 1 (Indicator functions) -For an indicator function, we require

Ω 1 A dµ := µ(A)

(which may be infinite).

We extend this linearly now for nonnegative functions which are sums of indicators: these functions are called simple functions.

Step 2 (Simple functions) -Let A 1 , . . . , A n be a finite collection of measurable sets. Let c 1 , . . . , c n be either nonnegative real numbers or +∞. Then we define

Ω n i=1 c i 1 A i dµ := n i=1 c i µ(A i ). If c i = ∞ and µ(A i ) = 0, we treat c i µ(A i ) = 0.

One can check the resulting sum does not depend on the representation of the simple function as c i 1 A i . In particular, it is compatible with the previous step.

Conveniently, this is already enough to define the integral for f : Ω → [0, +∞]. Note that [0, +∞] can be thought of as a topological space where we add new open sets (a, +∞] for each real number a to our usual basis of open intervals. Thus we can equip it with the Borel sigma-algebra. 1Step 3 (Nonnegative functions) -For each measurable function f : Ω → [0, +∞], let

Ω f dµ := sup 0≤s≤f Ω s dµ

where the supremum is taken over all simple s such that 0 ≤ s ≤ f . As before, this integral may be +∞.

One can check this is compatible with the previous definitions. At this point, we introduce an important term.

Definition 37. 1 

.2. A measurable (nonnegative) function

f : Ω → [0, +∞] is absolutely integrable or just integrable if Ω f dµ < ∞.

Warning: I find "integrable" to be really confusing terminology. Indeed, every measurable function from Ω to [0, +∞] can be assigned a Lebesgue integral, it's just that this integral may be +∞. So the definition is far more stringent than the name suggests. Even constant functions can fail to be integrable:

Example 37. 1.3 (We really should call it "finitely integrable")

The constant function 1 is not integrable on R, since R 1 dµ = µ(R) = +∞.

For this reason, I will usually prefer the term "integrable". (If it were up to me, I would call it "finitely integrable", and usually do so privately.)

Finally, this lets us integrate general functions.

Definition 37.1.4. In general, a measurable function

f : Ω → [-∞, ∞] is absolutely integrable or just integrable if |f | is.

Since we'll be using the first word, this is easy to remember: "absolutely integrable" requires taking absolute values.

Step 4 (Absolutely integrable functions) -If f : Ω → [-∞, ∞] is absolutely integrable, then we define

f + (x) = max {f (x), 0} f -(x) = min {f (x), 0}

and set

Ω f dµ = Ω |f + | dµ - Ω |f -| dµ which in particular is finite.

You may already start to see that we really like nonnegative functions: with the theory of measures, it is possible to integrate them, and it's even okay to throw in +∞'s everywhere. But once we start dealing with functions that can be either positive or negative, we have to start adding finiteness restrictions -actually essentially what we're doing is splitting the function into its positive and negative part, requiring both are finite, and then integrating.

To finish this section, we state for completeness some results that you probably could have guessed were true. Fix Ω = (Ω, A , µ), and let f and g be measurable real-valued functions such that f (x) = g(x) almost everywhere.

• (Almost-everywhere preservation) The function f is absolutely integrable if and only if g is, and if so, their Lebesgue integrals match.

• (Additivity) If f and g are absolutely integrable then

Ω f + g dµ = Ω f dµ + Ω g dµ.

The "absolutely integrable" hypothesis can be dropped if f and g are nonnegative.

• (Scaling) If f is absolutely integrable and c ∈ R then cf is absolutely integrable and

Ω cf dµ = c Ω f dµ.

The "absolutely integrable" hypothesis can be dropped if f is nonnegative and c > 0.

• (Monotoncity) If f and g are absolutely integrable and f ≤ g, then

Ω f dµ ≤ Ω g dµ.

The "absolutely integrable" hypothesis can be dropped if f and g are nonnegative.

There are more famous results like monotone/dominated convergence that are also true, but we won't state them here as we won't really have a use for them in the context of probability. (They appear later on in a bonus chapter.) §37.

2 Relation to Riemann integrals (or: actually computing Lebesgue integrals)

For closed intervals, this actually just works out of the box. f dµ.

Thus in practice, we do all theory with Lebesgue integrals (they're nicer), but when we actually need to compute [1, 4] x 2 dµ we just revert back to our usual antics with the Fundamental Theorem of Calculus.

Example 37.2.2 (Integrating x 2 over [1, 4]) Reprising our old example:

[1,4] x 2 dµ = 4 1 x 2 dx = 1 3 • 4 3 - 1 3 • 1 3 = 21.

This even works for improper integrals, if the functions are nonnegative. The statement is a bit cumbersome to write down, but here it is. 

f dµ = lim a ′ →a + b ′ →b - b ′ a ′ f (x) dx

where we allow both sides to be +∞ if f is not absolutely integrable.

The right-hand side makes sense since [a ′ , b ′ ] ⊊ (a, b) is a compact interval on which f is continuous. This means that improper Riemann integrals of nonnegative functions can just be regarded as Lebesgue ones over the corresponding open intervals.

It's probably better to just look at an example though.

Example 37.2.4 (Integrating 1/ √ x on (0, 1))

For example, you might be familiar with improper integrals like

1 0 1 √ x dx := lim ε→0 + 1 ε 1 √ x dx = lim ε→0 + 2 √ 1 -2 √ ε = 2.

(Note this appeared before as Problem 30C ⋆ .) In the Riemann integration situation, we needed the limit as ε → 0 + since otherwise 1

√

x is not defined as a function [0, 1] → R. However, it is a measurable nonnegative function (0, 1) → [0, +∞], and hence (0,1)

1 √ x dµ = 2.

If f is not nonnegative, then all bets are off. Indeed Problem 37B † is the famous counterexample. §37.3 A few harder problems to think about Problem 37A ⋆ (The indicator of the rationals). Take the indicator function 1 Q : R → {0, 1} ⊆ R for the rational numbers. 

f : (1, ∞) → R by f (x) = sin(x)

x . Show that f is not absolutely integrable, but that the improper Riemann integral 

integrals §38.1 Motivating limit interchange

Prototypical example for this section: 1 Q is good! One of the issues with the Riemann integral is that it behaves badly with respect to convergence of functions, and the Lebesgue integral deals with this. This is therefore often given as a poster child on why the Lebesgue integral has better behaviors than the Riemann one.

We technically have already seen this: consider the indicator function 1 Q , which is not Riemann integrable by Problem 37A ⋆ . But we can readily compute its Lebesgue integral over [0, 1], as

[0,1] 1 Q dµ = µ ([0, 1] ∩ Q) = 0 since it is countable.

This could be thought of as a failure of convergence for the Riemann integral.

Example 38.1.1 (1 Q is a limit of finitely supported functions)

We can define the sequence of functions g 1 , g 2 , . . . by

g n (x) = 1 (n!)

x is an integer 0 else.

Then each g n is piecewise continuous and hence Riemann integrable on [0, 1] (with integral zero), but lim n→∞ g n = 1 Q is not.

The limit here is defined in the following sense:

Definition 38.1.2. Let f and f 1 , f 2 , . . . : Ω → R be a sequence of functions. Suppose that for each ω ∈ Ω, the sequence

f 1 (ω), f 2 (ω), f 3 (ω), , . . .

converges to f (ω). Then we say f 1 , f 2 , . . . converges pointwise to the limit f , written lim n→∞ f n = f . We can define lim inf n→∞ f n and lim sup n→∞ f n similarly.

This is actually a fairly weak notion of convergence, for example: 

f (x) = 1 x = 0 0 otherwise.

This is why when thinking about the Riemann integral it is commonplace to work with stronger conditions like "uniformly convergent" and the like. However, with the Lebesgue integral, we can mostly not think about these! §38.

2 Overview

The three big-name results for exchanging pointwise limits with Lebesgue integrals is:

• Fatou's lemma: the most general statement possible, for any nonnegative measurable functions.

• Monotone convergence: "increasing limits" just work. 

Ω lim inf n→∞ f n dµ ≤ lim inf n→∞ Ω f n dµ .

Here we allow either side to be +∞.

Notice that there are no extra hypothesis on f n other than nonnegative: which makes this quite surprisingly versatile if you ever are trying to prove some general result.

§38.4 Everything else

The big surprise is how quickly all the "big-name" theorem follows from Fatou's lemma.

Here is the so-called "monotone convergence theorem".

Corollary 38.4.1 (Monotone convergence theorem)

Let f and f 1 , f 2 , . . . : Ω → [0, +∞] be a sequence of nonnegative measurable functions such that lim n f n = f and f n (ω) ≤ f (ω) for each n. Then f is measurable and

lim n→∞ Ω f n dµ = Ω f dµ.

Here we allow either side to be +∞.

Proof. We have

Ω f dµ = Ω lim inf n→∞ f n dµ ≤ lim inf n→∞ Ω f n dµ ≤ lim sup n→∞ Ω f n dµ ≤ Ω f dµ

where the first ≤ is by Fatou lemma, and the second by the fact that Ω f n ≤ Ω f for every n. This implies all the inequalities are equalities and we are done.

Remark 38.4.2 (The monotone convergence theorem does not require monotonicity!) -In the literature it is much more common to see the hypothesis Proof. There are three inequalities:

f 1 (ω) ≤ f 2 (ω) ≤ • • • ≤ f (ω) rather than just f n (ω) ≤ f (ω) for

(x) = 1 Q (x),

• The first inequality follows by Fatou on g + f n which is nonnegative.

• The second inequality is just lim inf ≤ lim sup. (This makes the theorem statement easy to remember!)

• The third inequality follows by Fatou on g -f n which is nonnegative. Then 

Ω f dµ = lim n→∞ Ω f n dµ . Proof. If f (ω) = lim n→∞ f n (ω), then f (ω) = lim inf n→∞ f n (ω) = lim sup n→∞ f n (ω)

n sin(n -1 x) √ x dx

then one starts by observing that the inner term is bounded by the absolutely integrable function x -1/2 . Therefore it equals (0,1)

lim n→∞ n sin(n -1 x) √ x dx = (0,1) x √ x dx = (0,1) √ x dx = 2 3 . §38

.5 Fubini and Tonelli §38.6 A few harder problems to think about problems 39 Bonus: A hint of Pontryagin duality

In this short chapter we will give statements about how to generalize our Fourier analysis (a bonus chapter Chapter 14) to a much wider class of groups G. §39.1 LCA groups Prototypical example for this section: T, R.

Earlier we played with R, which is nice because in addition to being a topological space, it is also an abelian group under addition. These sorts of objects which are both groups and spaces have a name.

Definition 39.1.1. A group G is a topological group is a Hausdorff1 topological space equipped also with a group operation (G, •), such that both maps

G × G → G by (x, y) → xy G → G by x → x -1 are continuous.

For our Fourier analysis, we need some additional conditions. Definition 39.1.2. A locally compact abelian (LCA) group G is one for which the group operation is abelian, and moreover the topology is locally compact: for every point p of G, there exists a compact subset K of G such that K ∋ p, and K contains some open neighborhood of p.

Our previous examples all fall into this category:

Example 39. 1 

.3 (Examples of locally compact abelian groups)

• Any finite group Z with the discrete topology is LCA.

• The circle group T is LCA and also in fact compact.

• The real numbers R are an example of an LCA group which is not compact.

These conditions turn out to be enough for us to define a measure on the space G. The relevant theorem, which we will just quote: Theorem 39. 1 

.4 (Haar measure)

Let G be a locally compact abelian group. We regard it as a measurable space using its Borel σ-algebra B(G). There exists a measure µ : B(G) → [0, ∞], called the Haar measure, satisfying the following properties:

• µ(gS) = µ(S) for every g ∈ G and measurable S. That means that µ is "translation-invariant" under translation by G.

• µ(K) is finite for any compact set K.

• if S is measurable, then µ(S) = inf {µ(U ) : U ⊇ S open}. • if U is open, then µ(U ) = sup {µ(S) : S ⊇ U measurable}.

Moreover, it is unique up to scaling by a positive constant.

Remark 39.1.5 -Note that if G is compact, then µ(G) is finite (and positive). For this reason the Haar measure on a LCA group G is usually normalized so µ(G) = 1.

For this chapter, we will only use the first two properties at all, and the other two are just mentioned for completeness. Note that this actually generalizes the chapter where we constructed a measure on B(R n ), since R n is an LCA group! So, in short: if we have an LCA group, we have a measure µ on it. • Z ∼ = T, since group homomorphisms Z → T are determined by the image of 1.

• T ∼ = Z. The characters are given by θ → nθ for n ∈ Z.

• R ∼ = R. This is because a nonzero continuous homomorphism R → S 1 is determined by the fiber above 1 ∈ S 1 . (Algebraic topologists might see covering projections here.)

• Z/nZ ∼ = Z/nZ, characters ξ being determined by the image ξ(1) ∈ T.

• G × H ∼ = G × H.

Exercise 39.2.3 ( Z ∼ = Z, for those who read Section 18.1). If Z is a finite abelian group, show that Z ∼ = Z, using the results of the previous example. You may now recognize that the bilinear form

• : Z × Z → T is exactly a choice of isomorphism Z → Z. It is not "canonical".

True to its name as the dual, and in analogy with (V ∨ ) ∨ ∼ = V for vector spaces V , we have: Theorem 39.2.4 (Pontryagin duality theorem)

For any LCA group G, there is an isomorphism

G ∼ = G by x → (ξ → ξ(x)) .

The compact case is especially nice.

Proposition 39.2.5 (G compact ⇐⇒ G discrete)

Let G be an LCA group. Then G is compact if and only if G is discrete.

Proof. Problem 39B.

§39.3 The orthonormal basis in the compact case

Let G be a compact LCA group, and work with its Haar measure. We may now let L 2 (G) be the space of square-integrable functions to C, i.e.

L 2 (G) = f : G → C such that G |f | 2 < ∞ .

Thus we can equip it with the inner form

⟨f, g⟩ = G f • g.

In that case, we get all the results we wanted before:

Theorem 39.3.1 (Characters of G form an orthonormal basis)

Assume G is LCA and compact (so G is discrete). Then the characters

(e ξ ) ξ∈ G by e ξ (x) = e(ξ(x)) = exp(2πiξ(x))

form an orthonormal basis of L 2 (G). Thus for each f ∈ L 2 (G) we have

f = ξ∈ G f (ξ)e ξ where f (ξ) = ⟨f, e ξ ⟩ = G f (x) exp(-2πiξ(x)) dµ.

The sum ξ∈ G makes sense since G is discrete. In particular,

• Letting G = Z for a finite group G gives "Fourier transform on finite groups".

• The special case G = Z/nZ has its own Wikipedia page: the "discrete-time Fourier transform".

• Letting G = T gives the "Fourier series" earlier.

§39.4 The Fourier transform of the non-compact case

If G is LCA but not compact, then Theorem 39.3.1 becomes false. On the other hand, it's still possible to define G. We can then try to write the Fourier coefficients anyways: let

f (ξ) = G f • e ξ dµ for ξ ∈ G and f : G → C.

The results are less fun in this case, but we still have, for example:

Theorem 39.4.1 (Fourier inversion formula in the non-compact case)

Let µ be a Haar measure on G. Then there exists a unique Haar measure ν on G (called the dual measure) such that: whenever f ∈ L 1 (G) and f ∈ L 1 ( G), we have

f (x) = G f (ξ)ξ(x) dν

for almost all x ∈ G (with respect to µ). If f is continuous, this holds for all x.

So while we don't have the niceness of a full inner product from before, we can still in some situations at least write f as integral in sort of the same way as before.

In particular, they have special names for a few special G:

• If G = R, then G = R, yielding the "(continuous) Fourier transform".

• If G = Z, then G = T, yielding the "discrete time Fourier transform".

§39.5 Summary

We summarize our various flavors of Fourier analysis from the previous sections in the following table. 

ξ ∈ Z ∼ = Z e(iξ • x) Discrete Fourier transform Z/nZ ξ ∈ Z/nZ e(ξx/n) Fourier series T ∼ = [-π, π] n ∈ Z exp(inx) Continuous Fourier transform R ξ ∈ R e(ξx) Discrete time Fourier transform Z ξ ∈ T ∼ = [-π, π] exp(iξn)

You might notice that the various names are awful. This is part of the reason I got confused as a high school student: every type of Fourier series above has its own Wikipedia article. If it were up to me, we would just use the term "G-Fourier transform", and that would make everyone's lives a lot easier. 

41

Large number laws (TO DO) 403 41.1 Notions of convergence . 

42

Stopped martingales (TO DO) 405 42.1 How to make money almost surely . 

N →∞ #{1 ≤ i ≤ N | a ≤ X i (ω) ≤ b} N = b -a ∀0 ≤ a < b ≤ 1

holds for almost all choices of ω.

Problem 40B (Side length of triangle independent from median). Let

X 1 , Y 1 , X 2 , Y 2 , X 3 , Y 3 be six independent standard Gaussians. Define triangle ABC in the Cartesian plane by A = (X 1 , Y 1 ), B = (X 2 , Y 2 ), C = (X 3 , Y 3 ).

Prove that the length of side BC is independent from the length of the A-median.

Let's write this measure-theoretically: we could consider Ω = A ⊔ B A = {ω for which world ends in 2012} B = {ω for which world does not end in 2012} .

We will assume that A and B are measurable sets, that is, A, B ∈ A . That means we could have good fun arguing about what the values of µ(A) and µ(B) should be ("a priori probability that the world ends in 2012"), but let's move on to a different silly example.

We will now introduce a new notion that we will need when we define conditional probabilities later.

Definition 42.2. 1. Let Ω = (Ω, A , µ) be a probability space. A sub-σ-algebra F on Ω is exactly what it sounds like: a σ-algebra F on the set Ω such that each A ∈ F is measurable (i.e., F ⊆ A ).

The motivation is that F is the σ-algebra of sets which let us ask questions about some piece of information. For example, in the 2012 example we gave above, we might take F = {∅, A, B, Ω}, which are the sets we care about if we are thinking only about 2012.

Here are some more serious examples.

Example 42.2.2 (Examples of sub-σ-algebras) (a) Let X : Ω → {0, 1, 2} be a random variable taking on one of three values. If we're interested in X then we could define

A = {ω : X(ω) = 1} B = {ω : X(ω) = 2} C = {ω : X(ω) = 3}

then we could write

F = {∅, A, B, C, A ∪ B, B ∪ C, C ∪ A, Ω} .

This is a sub-σ-algebra on F that lets us ask questions about X like "what is the probability X ̸ = 3", say.

(b) Now suppose Y : Ω → [0, 1] is another random variable. If we are interested in Y , the F that captures our curiosity is

F = {Y pre (B) | B ⊆ [0, 1] is measurable } .

You might notice a trend here which we formalize now:

Definition 42.2.3. Let X : Ω → R be a random variable. The sub-σ-algebra generated by X is defined by

σ(X) := {X pre (B) | B ⊆ R is measurable } .

If X 1 , . . . is a sequence (finite or infinite) of random variables, the sub-σ-algebra generated by them is the smallest σ-algebra which contains σ(X i ) for each i.

Finally, we can put a lot of these together -since we're talking about time, we learn more as we grow older, and this can be formalized. • and so on, with F n being the measurable sets "determined" only by the first n coin flips. We'll need the definition of conditional probability to define a martingale, but this turns out to be surprisingly tricky. Let's consider the following simple example to see why.

Definition 42.2.4. A filtration on

Ω = (Ω, A , µ) is a nested sequence 1 F 0 ⊆ F 1 ⊆ F 2 ⊆ . . .

Example 42.3.1 (Why high-school methods aren't enough here) Suppose we have two independent random variables X, Y distributed uniformly over [0, 1] (so we may as well take Ω = [0, 1] 2 ). We might try to ask the question:

"what is the expected value of X given that X + Y = 0.6"? Intuitively, we know the answer has to be 0.3. However, if we try to write down a definition, we quickly run into trouble. Ideally we want to say something like

E[X given X + Y = 0.6] = S X S 1

where

S = {ω ∈ Ω | X(ω) + Y (ω) = 0.6} .

The problem is that S is a set of measure zero, so we quickly run into 0 0 , meaning a definition of this shape will not work out.

The way that this is typically handled in measure theory is to use the notion of sub-σ-algebra that we defined. Let F be a sub-σ-algebra which captures the information This means we create a function assigning the "conditional expectation" to every point ω ∈ Ω, which is measurable with respect to F . Let X : Ω → R be an absolutely integrable random variable (meaning E[|X|] < ∞) over a probability space Ω, and let F be a sub-σ-algebra on it.

Then there exists a function η : Ω → R satisfying the following two properties:

• η is F -measurable (that is, measurable as a function (Ω, F , µ) → R); and

• for any set

A ∈ F we have E[η • 1 A ] = E[X • 1 A ].

Moreover, this random variable is unique up to almost sureness.

Proof.

Omitted, but relevant buzzword used is "Radon-Nikodym derivative".

Definition 42.3.3. Let η be as in the previous proposition.

• We denote η by E(X | F ) and call it the conditional expectation of X with respect to F . For this reason I try to be careful to remember to use parentheses rather than square brackets for conditional expectations; not everyone does this.

• If Y is a random variable then E(X | Y ) denotes E(X | σ(Y )), i.

Abuse of Notation 42.3.5. In addition, when we write Y = E(X | F ), there is some abuse of notation happening here since E(X | F ) is defined only up to some reasonable uniqueness (i.e. up to measure zero changes). So this really means that "Y satisfies the hypothesis of Proposition 42.3.2", but this is so pedantic that no one bothers.

properties

§42.4 Supermartingales

Prototypical example for this section: Visiting a casino is a supermartingale, assuming house odds.

Definition 42.4.1. Let X 0 , X 1 , . . . be a sequence of random variables on a probability space Ω, and let

F 0 ⊆ F 1 ⊆ • • • be a filtration.

Then (X n ) n≥0 is a supermartingale with respect to (F n ) n≥0 if the following conditions hold:

• X n is absolutely integrable for every n;

• X n is measurable with respect to F n ; and

• for each n = 1, 2, . . . the inequality

E(X n | F n-1 ) ≤ X n-1

holds for all ω ∈ Ω.

In a submartingale the inequality ≤ is replaced with ≥, and in a martingale it is replaced by =.

Abuse of Notation 42. 4 .2 (No one uses that filtration thing anyways). We will always take F n to be the σ-algebra generated by the previous variables X 0 , X 1 , . . . , X n-1 , and do so without further comment. Nonetheless, all the results that follow hold in the more general setting of a supermartingale with respect to some filtration.

We will prove all our theorems for supermartingales; the analogous versions for submartingales can be obtained by replacing ≤ with ≥ everywhere (since X n is a martingale iff -X n is a supermartingale) and for martingales by replacing ≤ with = everywhere (since X n is a martingale iff it is both a supermartingale and a submartingale).

Let's give examples.

Example 42.4.3 (Supermartingales) (a) Random walks: an ant starts at the position 0 on the number line. Every minute, it flips a fair coin and either walks one step left or one step right. If X t is the position at the tth time, then X t is a martingale, because

E(X t | X 0 , . . . , X t-1 ) = (X t-1 + 1) + (X t-1 -1) 2 = X t-1 .

(b) Casino game: Consider a gambler using the strategy described at the beginning of the chapter. This is a martingale, since every bet the gambler makes has expected value 0.

(c) Multiplying independent variables: Let X 1 , X 2 , . . . , be independent (not necessarily identically distributed) integrable random variables with mean 1.

Then the sequence Y 1 , Y 2 . . . defined by

Y n := X 1 X 2 . . . X n is a martingale; as E(Y n | Y 1 , . . . , Y n-1 ) = E[Y n ] • Y n-1 = Y n-1 .

(d) Iterated blackjack: Suppose one shows up to a casino and plays infinitely many games of blackjack. If X t is their wealth at time t, then X t is a supermartingale. This is because each game has negative expected value (house edge).

Example 42.4.4 (Frivolous/inflamamtory example -real life is a supermartingale) Let X t be your happiness on day t of your life. Life has its ups and downs, so it is not the case that X t ≤ X t-1 for every t. For example, you might win the lottery one day. However, on any given day, many things can go wrong (e.g. zombie apocalypse), and by Murphy's Law this is more likely than things going well. Also, as you get older, you have an increasing number of responsibilities and your health gradually begins to deteriorate. Thus it seems that E(X t | X 0 , . . . , X t-1 ) ≤ X t-1 is a reasonable description of the future -in expectation, each successive day is slightly worse than the previous one. ( In particular, if we set X t = -∞ on death, then as long as you have a positive probability of dying, the displayed inequality is obviously true.)

Before going on, we will state without proof one useful result: if a martingale is bounded, then it will almost certainly converge.

Theorem 42.4.5 (Doob's martingale convergence theorem) Let X 0 , . . . be a supermartingale on a probability space Ω such that sup

n∈Z ≥0 E [|X n |] < ∞.

Then, there exists a random variable

X ∞ : Ω → R such that X n a.s.

--→ X ∞ . §42.5 Optional stopping Prototypical example for this section: Las Vegas.

In the first section we described how to make money almost surely. The key advantage the gambler had was the ability to quit whenever he wanted (equivalently, an ability to control the size of the bets; betting $0 forever is the same as quitting.) Let's formalize a notion of stopping time.

The idea is we want to define a function τ : Ω → {0, 1, . . . , ∞} such that

• τ (ω) specifies the index after which we stop the martingale. Note that the decisions to stop after time n must be made with only the information available at that time -i.e., with respect to F n of the filtration.

• X τ ∧n is the random value representing the value at time n of the stopped martingale, where if n is after the stopping time, we just take it to be the our currently value after we leave.

So for example in a world ω where we stopped at time 3, then

X τ ∧0 (ω) = X 0 (ω), X τ ∧1 (ω) = X 1 (ω), X τ ∧2 (ω) = X 2 (ω), X τ ∧3 (ω) = X 3 (ω), but then X 3 (ω) = X τ ∧4 (ω) = X τ ∧5 (ω) = X τ ∧6 (ω) = . . .

since we have stopped -the value stops changing.

• X τ denotes the eventual value after we stop (or the limit X ∞ if we never stop).

Here's the compiled machine code.

Definition 42.5.1. Let F 0 ⊆ F 1 ⊆ • • • be a filtration on a probability space Ω.

• A stopping time is a function

τ : Ω → {0, 1, 2, . . . } ∪ {∞}

with the property that for each integer n, the set

{ω ∈ Ω | τ (ω) = n} is F n -measurable (i.e., is in F n ).

• For each n ≥ 0 we define X τ ∧n : Ω → R by

X τ ∧n (ω) = X min{τ (ω),n}• Finally, we let the eventual outcome be denoted by

X τ (ω) =        X τ (ω) (ω) τ (ω) ̸ = ∞ lim n→∞ X n (ω) τ (ω) = ∞ and lim n→∞ X n (ω) exists undefined otherwise.

We require that the "undefined" case occurs only for a set of measure zero (for example, if Theorem 42.4.5 applies). Otherwise we don't allow X τ to be defined.

Proposition 42.5.2 (Stopped supermartingales are still supermartingales)

Let X 0 , X 1 , . . . be a supermartingale. Then the sequence

X τ ∧0 , X τ ∧1 , . . .

is itself a supermartingale.

Proof. We have almost everywhere the inequalities

E (X τ ∧n | F n-1 ) = E X n-1 + 1 τ (ω)=n-1 (X n -X n-1 ) | F n-1 = E (X n-1 | F n-1 ) + E 1 τ (ω)=n-1 • (X n -X n-1 ) | F n-1 = X n-1 + 1 τ (ω)=n-1 • E (X n -X n-1 | F n-1 ) ≤ X n-1

as functions from Ω → R.

Theorem 42.5.3 (Doob's optional stopping theorem)

Let X 0 , X 1 , . . . be a supermartingale on a probability space Ω, with respect to a filtration F 0 ⊆ F 1 ⊆ • • • . Let τ be a stopping time with respect to this filtration. Suppose that any of the following hypotheses are true, for some constant C: (c) Finite bets: we have E[τ ] < ∞, and for each n ≥ 1, the conditional expectation

E (|X n -X n-1 | | F n )

takes on values at most C for almost all ω ∈ Ω satisfying τ (ω) ≥ n.

Then X τ is well-defined almost everywhere, and more importantly,

E[X τ ] ≤ E[X 0 ].

The last equation can be cheekily expressed as "the only winning move is not to play".

Proof.

do later tonight Exercise 42.5.4. Conclude that going to Las Vegas with the strategy described in the first section is a really bad idea. What goes wrong? §42.6 Fun applications of optional stopping (TO DO)

We now give three problems which showcase some of the power of the results we have developed so far. §42. 

Y n = (X 1 + • • • + X n ) 2 -n is a martingale.

(b) (de Moivre's martingale) Fix real numbers p and q such that p, q > 0 and p+q = 1.

Let X 1 , X 2 , . . . be i.i.d. random variables which equal +1 with probability p, and -1 with probability q. Show that

Y n = qp -1 X 1 +X 2 +•••+Xn

is a martingale.

(c) (Pólya's urn) An urn contains one red and one blue marble initially. Every minute, a marble is randomly removed from the urn, and two more marbles of the same color are added to the urn. Thus after n minutes, the urn will have n + 2 marbles. 

E[X 1 + X 2 + • • • + X τ ] = µE[τ ].

Problem 42D (Unbiased drunkard's walk). An ant starts at 0 on a number line, and walks left or right one unit with probability 1/2. It stops once it reaches either -17 or +8.

(a) Find the probability it reaches +8 before -17.

(b) Find the expected value of the amount of time it takes to reach either endpoint.

Problem 42E (Biased drunkard's walk). Let 0 < p < 1 be a real number 

44

Differential forms 425 44.1 Pictures of differential forms . 

45

Integrating differential forms 433 45.1 Motivation: line integrals . 

46

A bit of manifolds 441 46.1 Topological manifolds . 

As I have ranted about before, linear algebra is done wrong by the extensive use of matrices to obscure the structure of a linear map. Similar problems occur with multivariable calculus, so here I would like to set the record straight.

Since we are doing this chapter using morally correct linear algebra, it's imperative you're comfortable with linear maps, and in particular the dual space V ∨ which we will repeatedly use.

In this chapter, all vector spaces have norms and are finite-dimensional over R. So in particular every vector space is also a metric space (with metric given by the norm), and we can talk about open sets as usual. §43.1 The total derivative Prototypical example for this section:

If f (x, y) = x 2 + y 2 , then (Df ) (x,y) = 2xe ∨ 1 + 2ye ∨ 2 . First, let f : [a, b] → R.

You might recall from high school calculus that for every point p ∈ R, we defined f ′ (p) as the derivative at the point p (if it existed), which we interpreted as the slope of the "tangent line".

x p f ′ (p) That's fine, but I claim that the "better" way to interpret the derivative at that point is as a linear map, that is, as a function. If f ′ (p) = 1.5, then the derivative tells me that if I move ε away from p then I should expect f to change by about 1.5ε. In other words,

The derivative of f at p approximates f near p by a linear function.

What about more generally? Suppose I have a function like f : R 2 → R, say f (x, y) = x 2 + y 2 for concreteness or something. For a point p ∈ R 2 , the "derivative" of f at p ought to represent a linear map that approximates f at that point p. That means I want a linear map T : R 2 → R such that

f (p + v) ≈ f (p) + T (v) for small displacements v ∈ R 2 .

Even more generally, if f : U → W with U ⊆ V open (in the ∥•∥ V metric as usual), then the derivative at p ∈ U ought to be so that

f (p + v) ≈ f (p) + T (v) ∈ W.

(We need U open so that for small enough v, p + v ∈ U as well.) In fact this is exactly what we were doing earlier with f ′ (p) in high school.

p + v p

Image derived from [gk] The only difference is that, by an unfortunate coincidence, a linear map R → R can be represented by just its slope. And in the unending quest to make everything a number so that it can be AP tested, we immediately forgot all about what we were trying to do in the first place and just defined the derivative of f to be a number instead of a function.

The fundamental idea of Calculus is the local approximation of functions by linear functions. The derivative does exactly this.

Jean Dieudonné as quoted in [Pu02] continues:

In the classical teaching of Calculus, this idea is immediately obscured by the accidental fact that, on a one-dimensional vector space, there is a oneto-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined as a number instead of a linear form. This slavish subservience to the shibboleth of numerical interpretation at any cost becomes much worse . . . So let's do this right. The only thing that we have to do is say what "≈" means, and for this we use the norm of the vector space. 

: V → W such that lim ∥v∥ V →0 ∥f (p + v) -f (p) -T (v)∥ W ∥v∥ V = 0.

Then T is the total derivative of f at p. We denote this by (Df ) p , and say f is differentiable at p. If (Df ) p exists at every point, we say f is differentiable. 

→ R by v → 2a • e ∨ 1 (v) + 2b • e ∨ 2 (v).

Here, the notation e ∨ 1 and e ∨ 2 makes sense, because by definition (Df ) p ∈ V ∨ : these are functions from V to R! Let's check this manually with the limit definition. Set v = xe 1 + ye 2 , and note that the norm on V is ∥(x, y)∥ V = x 2 + y 2 while the norm on W is just the absolute value ∥c∥ W = |c|. Then we compute

∥f (p + v) -f (p) -T (v)∥ W ∥v∥ V = (a + x) 2 + (b + y) 2 -(a 2 + b 2 ) -(2ax + 2by) x 2 + y 2 = x 2 + y 2 x 2 + y 2 = x 2 + y 2 → 0

as ∥v∥ → 0. Thus, for p = ae 1 + be 2 we indeed have (Df

) p = 2a • e ∨ 1 + 2b • e ∨ 2 .

Remark 43.1.4 -As usual, differentiability implies continuity.

Remark 43. 1 .5 -Although U ⊆ V , it might be helpful to think of vectors from U and V as different types of objects (in particular, note that it's possible for 0 V / ∈ U ). The vectors in U are "inputs" on our space while the vectors coming from V are "small displacements". For this reason, I deliberately try to use p ∈ U and v ∈ V when possible.

§43.2 The projection principle

Before proceeding I need to say something really important.

Theorem 43.2.1 (Projection principle)

Let U be an open subset of the vector space V . Let W be an n-dimensional real vector space with basis w 1 , . . . , w n . Then there is a bijection between continuous functions f : U → W and n-tuples of continuous f 1 , f 2 , . . . , f n : U → R by projection onto the ith basis element, i.e.

f (v) = f 1 (v)w 1 + • • • + f n (v)w n .

Proof. Obvious.

The theorem remains true if one replaces "continuous" by "differentiable", "smooth", "arbitrary", or most other reasonable words. Translation:

To think about a function f : U → R n , it suffices to think about each coordinate separately.

For this reason, we'll most often be interested in functions f : U → R. That's why the dual space V ∨ is so important.

§43.3 Total and partial derivatives

Prototypical example for this section: If f (x, y) = x 2 + y 2 , then (Df ) : (x, y) → 2x • e ∨ 1 + 2y • e ∨ 2 , and ∂f ∂x = 2x, ∂f ∂y = 2y. Let U ⊆ V be open and let V have a basis e 1 , . . . , e n . Suppose f : U → R is a function which is differentiable everywhere, meaning (Df ) p ∈ V ∨ exists for every p. In that case, one can consider Df as itself a function:

Df : U → V ∨ p → (Df ) p .

This is a little crazy: to every point in U we associate a function in V ∨ . We say Df is the total derivative of f , to reflect how much information we're dealing with. We say (Df ) p is the total derivative at p.

Let's apply the projection principle now to Df . Since we picked a basis e 1 , . . . , e n of V , there is a corresponding dual basis e ∨ 1 , e ∨ 2 , . . . , e ∨ n . The Projection Principle tells us that Df can thus be thought of as just n functions, so we can write

Df = ψ 1 e ∨ 1 + • • • + ψ n e ∨ n .

In fact, we can even describe what the ψ i are.

Definition 43.3.1. The i th partial derivative of f : U → R, denoted

∂f ∂e i : U → R is defined by ∂f ∂e i (p) := lim t→0 f (p + te i ) -f (p) t .

You can think of it as "f ′ along e i ". From the above remarks, we can derive that

Df = ∂f ∂e 1 • e ∨ 1 + • • • + ∂f ∂e n • e ∨ n .

and so given a basis of V , we can think of Df as just the n partials.

Remark 43.3.4 -Keep in mind that each ∂f ∂e i is a function from U to the reals. That is to say,

(Df ) p = ∂f ∂e 1 (p) ∈R •e ∨ 1 + • • • + ∂f ∂e n (p) ∈R •e ∨ n ∈ V ∨ .

Example 43.3.5 (Partial derivatives of f (x, y) = x 2 + y 2 ) Let f : R 2 → R by (x, y) → x 2 + y 2 . Then in our new language,

Df : (x, y) → 2x • e ∨ 1 + 2y • e ∨ 2 .

Thus the partials are

∂f ∂x : (x, y) → 2x ∈ R and ∂f ∂y : (x, y) → 2y ∈ R

With all that said, I haven't really said much about how to find the total derivative itself. For example, if I told you f (x, y) = x sin y + x 2 y 4 you might want to be able to compute Df without going through that horrible limit definition I told you about earlier.

Fortunately, it turns out you already know how to compute partial derivatives, because you had to take AP Calculus at some point in your life. It turns out for most reasonable functions, this is all you'll ever need.

Theorem 43.3.6 (Continuous partials implies differentiable)

Let U ⊆ V be open and pick any basis e 1 , . . . , e n . Let f : U → R and suppose that ∂f ∂e i is defined for each i and moreover is continuous. Then f is differentiable and Df is given by

Df = n i=1 ∂f ∂e i • e ∨ i .

Proof. Not going to write out the details, but. The example f (x, y) = x 2 + y 2 is the same thing. That being said, who cares about x sin y + x 2 y 4 anyways? §43.4 (Optional) A word on higher derivatives Let U ⊆ V be open, and take f : U → W , so that Df : U → Hom(V, W ).

Well, Hom(V, W ) can also be thought of as a normed vector space in its own right: it turns out that one can define an operator norm on it by setting

∥T ∥ := sup ∥T (v)∥ W ∥v∥ V | v ̸ = 0 V .

So Hom(V, W ) can be thought of as a normed vector space as well. Thus it makes sense to write D(Df ) : U → Hom(V, Hom(V, W ))

which we abbreviate as D 2 f . Dropping all doubt and plunging on,

D 3 f : U → Hom(V, Hom(V, Hom(V, W ))).

I'm sorry. As consolation, we at least know that Hom(V, W ) ∼ = V ∨ ⊗ W in a natural way, so we can at least condense this to

D k f : V → (V ∨ ) ⊗k ⊗ W

rather than writing a bunch of Hom's.

Remark 43.4.1 -

If k = 2, W = R, then D 2 f (v) ∈ (V ∨ ) ⊗2

, so it can be represented as an n × n matrix, which for some reason is called a Hessian.

The most important property of the second derivative is that 

Theorem 43.4.2 (Symmetry of D 2 f ) Let f : U → W with U ⊆ V open. If (D 2 f ) p exists at some p ∈ U , then it is symmetric, meaning (D 2 f ) p (v 1 , v 2 ) = (D 2 f ) p (v 2 , v 1

(D k f ) p (v 1 , . . . , v k ) = (D k f ) p (v σ(1) , . . . , v σ(k) ).

In this chapter, all vector spaces are finite-dimensional real inner product spaces. We first start by (non-rigorously) drawing pictures of all the things that we will define in this chapter. Then we re-do everything again in its proper algebraic context.

§44.1 Pictures of differential forms

Before defining a differential form, we first draw some pictures. The key thing to keep in mind is

"The definition of a differential form is: something you can integrate." -Joe Harris

We'll assume that all functions are smooth, i.e. infinitely differentiable. Let U ⊆ V be an open set of a vector space V . Suppose that we have a function f : U → R, i.e. we assign a value to every point of U . Thus, if we specify a finite set S of points in U we can "integrate" over S by just adding up the values of the points:

0 + √ 2 + 3 + (-1) = 2 + √ 2.

So, a 0-form f lets us integrate over 0-dimensional "cells". But this is quite boring, because as we know we like to integrate over things like curves, not single points. So, by analogy, we want a 1-form to let us integrate over 1-dimensional cells: i.e. over curves. What information would we need to do that? To answer this, let's draw a picture of a curve c, which can be thought of as a function c : [0, 1] → U .

U c p v

We might think that we could get away with just specifying a number on every point of U (i.e. a 0-form f ), and then somehow "add up" all the values of f along the curve. We'll use this idea in a moment, but we can in fact do something more general. Notice how when we walk along a smooth curve, at every point p we also have some extra information: a tangent vector v. So, we can define a 1-form α as follows. A 0-form just took a point and gave a real number, but a 1-form will take both a point and a tangent vector at that point, and spit out a real number. So a 1-form α is a smooth function on pairs (p, v), where v is a tangent vector at p, to R. Hence

α : U × V → R.

Actually, for any point p, we will require that α(p, -) is a linear function in terms of the vectors: i.e. we want for example that α(p, 2v) = 2α(p, v). So it is more customary to think of α as:

Definition 44.1.2. A 1-form α is a smooth function α : U → V ∨ .

Like with Df , we'll use α p instead of α(p). So, at every point p, α p is some linear functional that eats tangent vectors at p, and spits out a real number. Thus, we think of α p as an element of

V ∨ ; α p ∈ V ∨ .

Next, we draw pictures of 2-forms. This should, for example, let us integrate over a blob (a so-called 2-cell) of the form c : [0, 1] × [0, 1] → U i.e. for example, a square in U . In the previous example with 1-forms, we looked at tangent vectors to the curve c. This time, at points we will look at pairs of tangent vectors in U : in the same sense that lots of tangent vectors approximate the entire curve, lots of tiny squares will approximate the big square in U .

U c p v w

So what should a 2-form β be? As before, it should start by taking a point p ∈ U , so β p is now a linear functional: but this time, it should be a linear map on two vectors v and w. Here v and w are not tangent so much as their span cuts out a small parallelogram. So, the right thing to do is in fact consider

β p ∈ V ∨ ∧ V ∨ .

That is, to use the wedge product to get a handle on the idea that v and w span a parallelogram. Another valid choice would have been (V ∧ V ) ∨ ; in fact, the two are isomorphic, but it will be more convenient to write it in the former.

Next question:

How can we build a 1-form from a 0-form?

Let f be a 0-form on U ; thus, we have a function f : U → R. Then in fact there is a very natural 1-form on U arising from f , appropriately called df . Namely, given a point p and a tangent vector v, the differential form (df ) p returns the change in f along v. In other words, it's just the total derivative (Df ) p (v).

U 3 √ 2 -1 0 √ 2 + ε v

Thus, df measures "the change in f ". Now, even if I haven't defined integration yet, given a curve c from a point a to b, what do you think c df should be equal to? Remember that df is the 1-form that measures "infinitesimal change in f ". So if we add up all the change in f along a path from a to b, then the answer we get should just be

c df = f (b) -f (a).

This is the first case of something we call Stokes' theorem.

Generalizing, how should we get from a 1-form to a 2-form? At each point p, the 2-form β gives a β p which takes in a "parallelogram" and returns a real number. Now suppose we have a 1-form α. Then along each of the edges of a parallelogram, with an appropriate sign convention the 1-form α gives us a real number. So, given a 1-form α, we define dα to be the 2-form that takes in a parallelogram spanned by v and w, and returns the measure of α along the boundary.

Now, what happens if you integrate dα along the entire square c? The right picture is that, if we think of each little square as making up the big square, then the adjacent boundaries cancel out, and all we are left is the main boundary. This is again just a case of the so-called Stokes' theorem. (c) Let V = R 3 with standard basis e 1 , e 2 , e 3 . Then a typical 2-form is given by

α p = f (p) • e ∨ 1 ∧ e ∨ 2 + g(p) • e ∨ 1 ∧ e ∨ 3 + h(p) • e ∨ 2 ∧ e ∨ 3 ∈ Λ 2 (V ∨ )

where f, g, h : V → R are smooth functions. Now, by the projection principle (Theorem 43.2.1) we only have to specify a function on each of n k basis elements of Λ k (V ∨ ). So, take any basis {e i } of V , and take the usual basis for Λ k (V ∨ ) of elements

e ∨ i 1 ∧ e ∨ i 2 ∧ • • • ∧ e ∨ i k .

Thus, a general k-form takes the shape

α p = 1≤i 1 <•••<i k ≤n f i 1 ,...,i k (p) • e ∨ i 1 ∧ e ∨ i 2 ∧ • • • ∧ e ∨ i k .

Since this is a huge nuisance to write, we will abbreviate this to just

α = I f I • de I

where we understand the sum runs over I = (i 1 , . 

∈ V ∨ and vectors v 1 , . . . , v k ∈ V , set (ξ 1 ∧ • • • ∧ ξ k )(v 1 , . . . , v k ) := det    ξ 1 (v 1 ) . . . ξ 1 (v k ) . . . . . . . . . ξ k (v 1 ) . . . ξ k (v k )    .

You can check that this is well-defined under e.g. v ∧ w = -w ∧ v and so on.

Example 44.3.4 (Evaluation of a differential form)

Set V = R 3 . Suppose that at some point p, the 2-form α returns

α p = 2e ∨ 1 ∧ e ∨ 2 + e ∨ 1 ∧ e ∨ 3 .

Let v 1 = 3e 1 + e 2 + 4e 3 and v 2 = 8e 1 + 9e 2 + 5e 3 . Then

α p (v 1 , v 2 ) = 2 det 3 8 1 9 + det 3 8 4 5 = 21.

What does this definition mean? One way to say it is that

If I walk to a point p ∈ U , a k-form α will take in k vectors v 1 , . . . , v k and spit out a number, which is to be interpreted as a (signed) volume.

Picture:

U p v 1 v 2 α p (v 1 , v 2 ) ∈ R

In other words, at every point p, we get a function α p . Then I can feed in k vectors to α p and get a number, which I interpret as a signed volume of the parallelepiped spanned by the {v i }'s in some way (e.g. the flux of a force field). That's why α p as a "function" is contrived to lie in the wedge product: this ensures that the notion of "volume" makes sense, so that for example, the equality

α p (v 1 , v 2 ) = -α p (v 2 , v 1 ) holds.

This is what makes differential forms so fit for integration.

§44.4 Exterior derivatives

Prototypical example for this section: Possibly dx 1 = e ∨ 1 . We now define the exterior derivative df that we gave pictures of at the beginning of the chapter. It turns out that the exterior derivative is easy to compute given explicit coordinates to work with.

Firstly, we define the exterior derivative of a function f : U → R, as

df := Df = i ∂f ∂e i e ∨ i

In particular, suppose V = R n and f (x 1 , . Abuse of Notation 44.4.2. Unfortunately, someone somewhere decided it would be a good idea to use "x 1 " to denote e ∨ 1 (because obviously 1 x 1 means "the function that takes (x 1 , . . . , x n ) ∈ R n to x 1 ") and then decided that

dx 1 := e ∨ 1 .

This notation is so entrenched that I have no choice but to grudgingly accept it. Note that it's not even right, since technically it's (dx 1 ) p = e ∨ 1 ; dx 1 is a 1-form. Next, we can evaluate d(df ) as prescribed: it is

d 2 f = (12x 2 dx + 2dz) ∧ dx + (6y dy) ∧ dy + 2(dx ∧ dz) = 12x 2 (dx ∧ dx) + 2(dz ∧ dx) + 6y(dy ∧ dy) + 2(dx ∧ dz) = 2(dz ∧ dx) + 2(dx ∧ dz) = 0.

So surprisingly, d 2 f is the zero map. Here, we have exploited Abuse of Notation 44.4.2 for the first time, in writing dx, dy, dz.

And in fact, this is always true in general:

Theorem 44.4.6 (Exterior derivative vanishes)

Let α be any k-form. Then d 2 (α) = 0. Even more succinctly,

d 2 = 0.

The proof is left as Problem 44B.

1 Sarcasm.

Exercise 44.4.7. Compare the statement d 2 = 0 to the geometric picture of a 2-form given at the beginning of this chapter. Why does this intuitively make sense?

Here are some other properties of d:

• As we just saw, d 2 = 0.

• For a k-form α and ℓ-form β, one can show that

d(α ∧ β) = dα ∧ β + (-1) k (α ∧ dβ). • If f : U → R is smooth, then df = Df .

In fact, one can show that df as defined above is the unique map sending k-forms to (k + 1)-forms with these properties. So, one way to define df is to take as axioms the bulleted properties above and then declare d to be the unique solution to this functional equation. In any case, this tells us that our definition of d does not depend on the basis chosen.

Recall that df measures the change in boundary. In that sense, d 2 = 0 is saying something like "the boundary of the boundary is empty". We'll make this precise when we see Stokes' theorem in the next chapter.

§44.5 Closed and exact forms

Let α be a k-form. A natural question arises: are there closed forms which are not exact? Surprisingly, the answer to this question is tied to topology. Here is one important example.

Example 44.5.4 (The angle form)

Let U = R 2 \ {0}, and let θ(p) be the angle formed by the x-axis and the line from the origin to p.

The 1-form α : U → (R 2 ) ∨ defined by α = -y dx + x dy x 2 + y 2 is called the angle form: given p ∈ U it measures the change in angle θ(p) along a tangent vector. So intuitively, "α = dθ". Indeed, one can check directly that the angle form is closed. However, α is not exact: there is no global smooth function θ : U → R having α as a derivative. This reflects the fact that one can actually perform a full 2π rotation around the origin, i.e. θ only makes sense mod 2π. Thus existence of the angle form α reflects the possibility of "winding" around the origin.

So the key idea is that the failure of a closed form to be exact corresponds quite well with "holes" in the space: the same information that homotopy and homology groups are trying to capture. To draw another analogy, in complex analysis Cauchy-Goursat only works when U is simply connected. The "hole" in U is being detected by the existence of a form α. The so-called de Rham cohomology will make this relation explicit. Now that definition was a pain to write, so we will define a differential 1-form c * α on [a, b] to swallow that entire thing: specifically, in this case we define c * α to be

(c * α) t (ε) = α c(t) • (Dc) t (ε)

(here ε is some displacement in time). Thus, we can more succinctly write

c α := [a,b] c * α.

This is a special case of a pullback: roughly, if ϕ : 1we can resort to our old definition of an integral. Let's now do this in full generality.

U → U ′ (where U ⊆ V , U ′ ⊆ V ′ ), we can change any differential k-form α on U ′ to a k-form on U . In particular, if U = [a, b],

§45.2 Pullbacks

Let V and V ′ be finite dimensional real vector spaces (possibly different dimensions) and suppose U and U ′ are open subsets of each; next, consider a k-form α on U ′ .

Given a map ϕ : U → U ′ we now want to define a pullback in much the same way as before. Picture:

U p U ′ q = φ(p) φ α α q (. . . ) ∈ R φ * α

Well, there's a total of about one thing we can do. Specifically: α accepts a point in U ′ and k tangent vectors in V ′ , and returns a real number. We want ϕ * α to accept a point in p ∈ U and k tangent vectors v 1 , . . . , v k in V , and feed the corresponding information to α.

Clearly we give the point q = ϕ(p). As for the tangent vectors, since we are interested in volume, we take the derivative of ϕ at p, (Dϕ) p , which will scale each of our vectors v i into some vector in the target V ′ . To cut a long story short: Definition 45.2.1. Given ϕ : U → U ′ and α a k-form, we define the pullback

(ϕ * α) p (v 1 , . . . , v k ) := α ϕ(p) ((Dϕ) p (v 1 ), . .

. , (Dϕ) p (v k )) .

There is a more concrete way to define the pullback using bases. Suppose w 1 , . . . , w n is a basis of V ′ and e 1 , . . . , e m is a basis of V . Thus, by the projection principle (Theorem 43.2.1) the map ϕ : V → V ′ can be thought of as

ϕ(v) = ϕ 1 (v)w 1 + • • • + ϕ n (v)w n

where each ϕ i takes in a v ∈ V and returns a real number. We know also that α can be written concretely as α = J⊆{1,...,n}

f J w J .

Then, we define ϕ * α = I⊆{1,...,m}

(f I • ϕ)(Dϕ i 1 ∧ • • • ∧ Dϕ i k ).

A diligent reader can check these definitions are equivalent.

Example 45.2.2 (Computation of a pullback)

Let V = R 2 with basis e 1 and e 2 , and suppose ϕ : V → V ′ is given by sending

ϕ(ae 1 + be 2 ) = (a 2 + b 2 )w 1 + log(a 2 + 1)w 2 + b 3 w 3

where w 1 , w 2 , w 3 is a basis for V ′ . Consider the form α q = f (q)w 1 ∧ w 3 , where

f : V ′ → R. Then (ϕ * α) p = f (ϕ(p)) • (2ae ∨ 1 + 2be ∨ 2 ) ∧ (3b 2 e ∨ 2 ) = f (ϕ(p)) • 6ab 2 • e ∨ 1 ∧ e ∨ 2 .

It turns out that the pullback basically behaves nicely as possible, e.g.

• ϕ * (cα + β) = cϕ * α + ϕ * β (linearity) • ϕ * (α ∧ β) = (ϕ * α) ∧ (ϕ * β) • ϕ * 1 (ϕ * 2 (α)) = (ϕ 2 • ϕ 1 ) * (α) (naturality)

but I won't take the time to check these here (one can verify them all by expanding with a basis).

§45.3 Cells

Prototypical example for this section: A disk in R 2 can be thought of as the cell [0, R] × [0, 2π] → R 2 by (r, θ) → (r cos θ)e 1 + (r sin θ)e 2 .

Now that we have the notion of a pullback, we can define the notion of an integral for more general spaces. Specifically, to generalize the notion of integrals we had before: Since c * α is a k-form on the k-dimensional unit box, it can be written as f (x 1 , . . . , x n ) dx 1 ∧ • • • ∧ dx n , So the above integral could also be written as Here's some geometric intuition for what's happening. Given a k-cell in V , a differential k-form α accepts a point p and some tangent vectors v 1 , . . . , v k and spits out a number α p (v 1 , . . . , v k ), which as before we view as a signed hypervolume. Then the integral adds up all these infinitesimals across the entire cell. In particular, if V = R k and we take the form α :

Definition 45.3.1. A k-cell is a smooth function c : [a 1 , b 1 ] × [a 2 , b 2 ] × . . . [a k , b k ] → V .

1 0 • • • 1 0 f (x 1 , . . . , x n ) dx 1 ∧ • • • ∧ dx n .

p → e ∨ 1 ∧ • • • ∧ e ∨ k ,

then what these α's give is the kth hypervolume of the cell. For this reason, this α is called the volume form on R k .

You'll notice I'm starting to play loose with the term "cell": while the cell c : [0, R] × [0, 2π] → R 2 is supposed to be a function I have been telling you to think of it as a unit disk (i.e. in terms of its image). In the same vein, a curve [0, 1] → V should be thought of as a curve in space, rather than a function on time.

This error turns out to be benign. Let α be a k-form on U and c :

[a 1 , b 1 ] × • • • × [a k , b k ] → U a k-cell. Suppose ϕ : [a ′ 1 , b ′ 1 ] × . . . [a ′ k , b ′ k ] → [a 1 , b 1 ] × • • • × [a k , b k ]; it is a reparametrization if ϕ is bijective and (Dϕ) p is always invertible (think "change of variables"); thus c • ϕ : [a ′ 1 , b ′ 1 ] × • • • × [a ′ k , b ′ k ] → U is a k-cell as well.

Then it is said to preserve orientation if det(Dϕ) p > 0 for all p and reverse orientation if det(Dϕ) p < 0 for all p. Proof. Use naturality of the pullback to reduce it to the corresponding theorem in normal calculus. where each a i ∈ R and c i is a k-cell. We define c α = i a i c i .

So for example, if we had parametrized the unit circle as

[0, 1] × [0, 1] → R 2 by (r, t) → R cos(2πt)e 1 + R sin(2πt)e 2 ,

In particular, a 0-chain consists of several points, each with a given weight. Now, how do we define the boundary? For a 1-cell [a, b] → U , as I hinted earlier we want the answer to be the 0-chain {c(b)} -{c(a)}. Here's how we do it in general.

Definition 45.4.2. Suppose

c : [0, 1] k → U is a k-cell. Then the boundary of c, denoted ∂c : [0, 1] k-1 → U , is the (k -1)-chain defined as follows. For each i = 1, . . . , k define (k -1)-chains by c start i : (t 1 , . . . , t k-1 ) → c(t 1 , . . . , t i-1 , 0, t i , . . . , t k ) c stop i : (t 1 , . . . , t k-1 ) → c(t 1 , . . . , t i-1 , 1, t i , . . . , t k ).

Then

∂c := k i=1 (-1) i+1 c stop i -c start i .

Finally, the boundary of a chain is the sum of the boundaries of each cell (with the appropriate weights). That is, ∂( a i c i ) = a i ∂c i .

Question 45.4.3. Satisfy yourself that one can extend this definition to a k-cell c defined on

c : [a 1 , b 1 ] × • • • × [a k , b k ] → V (rather than from [0, 1] k → V ).

Example 45.4.4 (Examples of boundaries)

Consider the 2-cell c : [0, 1] 2 → R 2 shown below.

[0, 1] 2 c p 1 p 2 p 3 p 4 c

Here p 1 , p 2 , p 3 , p 4 are the images of (0, 0), (0, 1), (1, 1), (1, 0), respectively. Formally, ∂c is given by

∂c = ((t → c(1, t)) -(t → c(0, t))) -((t → c(t, 1)) -(t → c(t, 0))) .

I apologize for the eyesore notation caused by inline functions. Let's make amends and just write ∂c

= [p 2 , p 3 ] -[p 1 , p 4 ] -[p 4 , p 3 ] + [p 1 , p 2 ]

where each "interval" represents the 1-cell shown by the reddish arrows on the right, after accounting for the minus signs. We can take the boundary of this as well, and obtain an empty chain as

∂(∂c) = 4 i=1 {p i+1 } -{p i } = 0.

Example 45.4.5 (Boundary of a unit disk)

Consider the unit disk given by

c : [0, 1] × [0, 2π] → R 2 by (r, θ) → s cos(2πt)e 1 + s sin(2πt)e 2 .

The four parts of the boundary are shown in the picture below:

r θ [0, 1] 2 c

Note that two of the arrows more or less cancel each other out when they are integrated. Moreover, we interestingly have a degenerate 1-cell at the center of the circle; it is a constant function [0, 1] → R 2 which always gives the origin.

Obligatory theorem, analogous to d 2 = 0 and left as a problem. We now have all the ingredients to state Stokes' theorem for cells.

Theorem 45.5.1 (Stokes' theorem for cells)

Take U ⊆ V as usual, let c : [0, 1] k → U be a k-cell and let α : U → Λ k-1 (V ∨ ) be a (k -1)-form. Then c dα = ∂c α.

In particular, if dα = 0 then the left-hand side vanishes. Problem 45B. Show that ∂ 2 = 0.

Problem 45C (Pullback and d commute). Let U and U ′ be open sets of vector spaces V and V ′ and let ϕ : U → U ′ be a smooth map between them. Prove that for any differential form α on U ′ we have ϕ * (dα) = d(ϕ * α).

Problem 45D (Arc length isn't a form). Show that there does not exist a 1-form α on R 2 such that for a curve c : [0, 1] → R 2 , the integral c α gives the arc length of c.

Problem 45E.

An exact k-form α is one satisfying α = dβ for some β. Prove that

C 1 α = C 2 α

where C 1 and C 2 are any concentric circles in the plane and α is some exact 1-form.

Last chapter, we stated Stokes' theorem for cells. It turns out there is a much larger class of spaces, the so-called smooth manifolds, for which this makes sense.

Unfortunately, the definition of a smooth manifold is complete garbage, and so by the time I am done defining differential forms and orientations, I will be too lazy to actually define what the integral on it is, and just wave my hands and state Stokes' theorem.

§46.1 Topological manifolds

Prototypical example for this section: S 2 : "the Earth looks flat".

Long ago, people thought the Earth was flat, i.e. homeomorphic to a plane, and in particular they thought that π 2 (Earth) = 0. But in fact, as most of us know, the Earth is actually a sphere, which is not contractible and in particular π 2 (Earth) ∼ = Z. This observation underlies the definition of a manifold:

An n-manifold is a space which locally looks like R n .

Actually there are two ways to think about a topological manifold M :

• "Locally": at every point p ∈ M , some open neighborhood of p looks like an open set of R n . For example, to someone standing on the surface of the Earth, the Earth looks much like R 2 .

• "Globally": there exists an open cover of M by open sets {U i } i (possibly infinite) such that each U i is homeomorphic to some open subset of R n . For example, from outer space, the Earth can be covered by two hemispherical pancakes. 

ϕ i : U i ∼ = -→ E i ⊆ R n

where each E i is an open subset of R n . Each ϕ i : U i → E i is called a chart, and together they form a so-called atlas.

Remark 46.1.3 -Here "E" stands for "Euclidean". I think this notation is not standard; usually people just write ϕ i (U i ) instead.

Remark 46.1.4 -This definition is nice because it doesn't depend on embeddings: a manifold is an intrinsic space M , rather than a subset of R N for some N . Analogy: an abstract group G is an intrinsic object rather than a subgroup of S n .

Example 46.1.5 (An atlas on S 1 )

Here is a picture of an atlas for S 1 , with two open sets.

S 1 U 2 U 1 E 1 φ 1 E 2 φ 2

Question 46.1.6. Where do you think the words "chart" and "atlas" come from?

Example 46. 1.7 (Some examples of topological manifolds) (a) As discussed at length, the sphere S 2 is a 2-manifold: every point in the sphere has a small open neighborhood that looks like D 2 . One can cover the Earth with just two hemispheres, and each hemisphere is homeomorphic to a disk. Let M be a topological n-manifold with atlas

{U i ϕ i -→ E i }.

Definition 46.2.1. For any i, j such that U i ∩ U j ̸ = ∅, the transition map ϕ ij is the composed map

ϕ ij : E i ∩ ϕ img i (U i ∩ U j ) ϕ -1 i --→ U i ∩ U j ϕ j -→ E j ∩ ϕ img j (U i ∩ U j ).

Sorry for the dense notation, let me explain. The intersection with the image ϕ img i (U i ∩ U j ) and the image ϕ img j (U i ∩ U j ) is a notational annoyance to make the map well-defined and a homeomorphism. The transition map is just the natural way to go from E i → E j , restricted to overlaps. Picture below, where the intersections are just the green portions of each E 1 and E 2 :

S 1 U 2 U 1 E 1 φ 1 E 2 φ 2 φ 12

We want to add enough structure so that we can use differential forms. Definition 46.2.2. We say M is a smooth manifold if all its transition maps are smooth.

This definition makes sense, because we know what it means for a map between two open sets of R n to be differentiable.

With smooth manifolds we can try to port over definitions that we built for R n onto our manifolds. So in general, all definitions involving smooth manifolds will reduce to something on each of the coordinate charts, with a compatibility condition.

As an example, here is the definition of a "smooth map": 

E i ϕ -1 i --→ U i → M f -→ R is smooth as a function E i → R.

(b) Let M and N be smooth with atlases

{U M i ϕ i -→ E M i } i and {U N j ϕ j

-→ E N j } j , A map f : M → N is smooth if for every i and j, the composed map

E i ϕ -1 i --→ U i → M f -→ N ↠ U j ϕ j -→ E j is smooth, as a function E i → E j .

§46.3 Regular value theorem

Prototypical example for this section: x 2 + y 2 = 1 is a circle! Despite all that I've written about general manifolds, it would be sort of mean if I left you here because I have not really told you how to actually construct manifolds in practice, even though we know the circle x 2 + y 2 = 1 is a great example of a one-dimensional manifold embedded in R 2 .

Theorem 46.3.1 (Regular value theorem)

Let V be an n-dimensional real normed vector space, let U ⊆ V be open and let f 1 , . . . , f m : U → R be smooth functions. Let M be the set of points p ∈ U such that

f 1 (p) = • • • = f m (p) = 0.

Assume M is nonempty and that the map

V → R m by v → ((Df 1 ) p (v), . . . , (Df m ) p (v))

has rank m, for every point p ∈ M . Then M is a manifold of dimension n -m.

For a proof, see [Sj05, Theorem 6.3].

One very common special case is to take m = 1 above.

Corollary 46.3.2 (Level hypersurfaces)

Let V be a finite-dimensional real normed vector space, let U ⊆ V be open and let f : U → R be smooth. Let M be the set of points p ∈ U such that f (p) = 0.

If M ̸ = ∅ and (Df ) p is not the zero map for any p ∈ M , then M is a manifold of dimension dim V -1.

Example 46.3.3 (The circle

x 2 + y 2 -c = 0) Let f (x, y) = x 2 + y 2 -c, f : R 2 → R,

where c is a positive real number. Note that

Df = 2x • dx + 2y • dy

which in particular is nonzero as long as (x, y) ̸ = (0, 0), i.e. as long as c ̸ = 0. Thus:

• When c > 0, the resulting curve -a circle with radius √ c -is a onedimensional manifold, as we knew.

• When c = 0, the result fails. Indeed, M is a single point, which is actually a zero-dimensional manifold!

We won't give further examples since I'm only mentioning this in passing in order to increase your capacity to write real concrete examples. (But [Sj05, Chapter 6.2] has some more examples, beautifully illustrated.)

§46.4 Differential forms on manifolds

We already know what a differential form is on an open set U ⊆ R n . So, we naturally try to port over the definition of differentiable form on each subset, plus a compatibility condition.

Let M be a smooth manifold with atlas

{U i ϕ i -→ E i } i .

Definition 46.4.1. A differential k-form α on a smooth manifold M is a collection {α i } i of differential k-forms on each E i , such that for any j and i we have that

α j = ϕ * ij (α i ).

In English: we specify a differential form on each chart, which is compatible under pullbacks of the transition maps.

Prototypical example for this section: Left versus right, clockwise vs. counterclockwise.

This still isn't enough to integrate on manifolds. We need one more definition: that of an orientation.

The main issue is the observation from standard calculus that

b a f (x) dx = - a b f (x) dx.

Consider then a space M which is homeomorphic to an interval. If we have a 1-form α, how do we integrate it over M ? Since M is just a topological space (rather than a subset of R), there is no default "left" or "right" that we can pick. As another example, if M = S 1 is a circle, there is no default "clockwise" or "counterclockwise" unless we decide to embed M into R 2 .

To work around this we have to actually have to make additional assumptions about our manifold.

Definition 46.5.1. A smooth n-manifold is orientable if there exists a differential n-form ω on M such that for every p ∈ M ,

ω p ̸ = 0.

Recall here that ω p is an element of Λ n (V ∨ ). In that case we say ω is a volume form of M .

How do we picture this definition? If we recall that an differential form is supposed to take tangent vectors of M and return real numbers. To this end, we can think of each point p ∈ M as having a tangent plane T p (M ) which is n-dimensional. Now since the volume form ω is n-dimensional, it takes an entire basis of the T p (M ) and gives a real number. So a manifold is orientable if there exists a consistent choice of sign for the basis of tangent vectors at every point of the manifold.

For "embedded manifolds", this just amounts to being able to pick a nonzero field of normal vectors to each point p ∈ M . For example, S 1 is orientable in this way.

S 1

Similarly, one can orient a sphere S 2 by having a field of vectors pointing away (or towards) the center. This is all non-rigorous, because I haven't defined the tangent plane T p (M ); since M is in general an intrinsic object one has to be quite roundabout to define T p (M ) (although I do so in an optional section later). In any event, the point is that guesses about the orientability of spaces are likely to be correct. 

6 Stokes' theorem for manifolds

Stokes' theorem in the general case is based on the idea of a manifold with boundary M , which I won't define, other than to say its boundary ∂M is an n -1 dimensional manifold, and that it is oriented if M is oriented. An example is M = D 2 , which has boundary ∂M = S 1 .

Next, Definition 46.6.1. The support of a differential form α on M is the closure of the set

{p ∈ M | α p ̸ = 0} .

If this support is compact as a topological space, we say α is compactly supported.

Remark 46.6.2 -For example, volume forms are supported on all of M . Now, one can define integration on oriented manifolds, but I won't define this because the definition is truly awful. Then Stokes' theorem says Theorem 46. 6 

.3 (Stokes' theorem for manifolds)

Let M be a smooth oriented n-manifold with boundary and let α be a compactly supported (n -1)-form. Then

M dα = ∂M α.

All the omitted details are developed in full in [Sj05] . §46.7 (Optional) The tangent and cotangent space Prototypical example for this section: Draw a line tangent to a circle, or a plane tangent to a sphere.

Let M be a smooth manifold and p ∈ M a point. I omitted the definition of T p (M ) earlier, but want to actually define it now.

As I said, geometrically we know what this should look like for our usual examples. For example, if M = S 1 is a circle embedded in R 2 , then the tangent vector at a point p should just look like a vector running off tangent to the circle. Similarly, given a sphere M = S 2 , the tangent space at a point p along the sphere would look like plane tangent to M at p.

S 1 T p (M ) v ∈ T p (M )

p However, one of the points of all this manifold stuff is that we really want to see the manifold as an intrinsic object, in its own right, rather than as embedded in R n . 1 So, we would like our notion of a tangent vector to not refer to an ambient space, but only to intrinsic properties of the manifold M in question.

§46.7.i Tangent space

To motivate this construction, let us start with an embedded case for which we know the answer already: a sphere.

Suppose f : S 2 → R is a function on a sphere, and take a point p. Near the point p, f looks like a function on some open neighborhood of the origin. Thus we can think of taking a directional derivative along a vector ⃗ v in the imagined tangent plane (i.e. some partial derivative). For a fixed ⃗ v this partial derivative is a linear map

D ⃗ v : C ∞ (M ) → R.

It turns out this goes the other way: if you know what D ⃗ v does to every smooth function, then you can recover v. This is the trick we use in order to create the tangent space. Rather than trying to specify a vector ⃗ v directly (which we can't do because we don't have an ambient space),

The vectors are partial-derivative-like maps.

More formally, we have the following. Definition 46.7.1. A derivation D at p is a linear map D : C ∞ (M ) → R (i.e. assigning a real number to every smooth f ) satisfying the following Leibniz rule: for any f , g we have the equality

D(f g) = f (p) • D(g) + g(p) • D(f ) ∈ R.

This is just a "product rule". Then the tangent space is easy to define: Definition 46.7.2. A tangent vector is just a derivation at p, and the tangent space T p (M ) is simply the set of all these tangent vectors.

In this way we have constructed the tangent space. 1 This can be thought of as analogous to the way that we think of a group as an abstract object in its own right, even though Cayley's Theorem tells us that any group is a subgroup of the permutation group. Note this wasn't always the case! During the 19th century, a group was literally defined as a subset of GL(n) or of Sn. In fact Sylow developed his theorems without the word "group" Only much later did the abstract definition of a group was given, an abstract set G which was independent of any embedding into Sn, and an object in its own right.

§46.7.ii The cotangent space

In fact, one can show that the product rule for D is equivalent to the following three conditions:

1. D is linear, meaning D(af + bg) = aD(f ) + bD(g).

2. D(1

M ) = 0

, where 1 M is the constant function on M .

3. D(f g) = 0 whenever f (p) = g(p) = 0. Intuitively, this means that if a function h = f g vanishes to second order at p, then its derivative along D should be zero.

This suggests a third equivalent definition: suppose we define

m p := {f ∈ C ∞ M | f (p) = 0}

to be the set of functions which vanish at p (this is called the maximal ideal at p). In that case,

m 2 p = i f i • g i | f i (p) = g i (p) = 0

is the set of functions vanishing to second order at p. Thus, a tangent vector is really just a linear map m p /m 2 p → R.

In other words, the tangent space is actually the dual space of m p /m 2 p ; for this reason, the space m p /m 2 p is defined as the cotangent space (the dual of the tangent space). This definition is even more abstract than the one with derivations above, but has some nice properties:

• it is coordinate-free, and • it's defined only in terms of the smooth functions M → R, which will be really helpful later on in algebraic geometry when we have varieties or schemes and can repeat this definition.

§46.7.iii Sanity check

With all these equivalent definitions, the last thing I should do is check that this definition of tangent space actually gives a vector space of dimension n. 

m 0 = {smooth functions f : f (0) = 0} m 2 0 = {smooth functions f : f (0) = 0, (∇f ) 0 = 0}.

In other words m 2 0 is the set of functions which vanish at 0 and such that all first derivatives of f vanish at zero. Thus, it follows that there is an isomorphism

m 0 /m 2 0 ∼ = R n by f → ∂f ∂x 1 (0), . . . , ∂f ∂x n (0)

and so the cotangent space, hence tangent space, indeed has dimension n. §46.8 A few harder problems to think about Problem 46A. Show that a differential 0-form on a smooth manifold M is the same thing as a smooth function M → R.

some applications of regular value theorem here

XIII Algebraic NT I: Rings of Integers Part XIII: Contents

Here's a first taste of algebraic number theory. This is really close to the border between olympiads and higher math. You've always known that a+ √ 2b had a "norm" a 2 -2b 2 , and that somehow this norm was multiplicative. You've also always known that roots come in conjugate pairs. You might have heard of minimal polynomials but not know much about them.

This chapter and the next one will make all these vague notions precise. It's drawn largely from the first chapter of [Og10] .

§47.1 Motivation from high school algebra

This is adapted from my blog, Power Overwhelming 1 .

In high school precalculus, you'll often be asked to find the roots of some polynomial with integer coefficients. For instance,

x 3 -x 2 -x -15 = (x -3)(x 2 + 2x + 5)

has roots 3, -1 + 2i, -1 -2i. Or as another example,

x 3 -3x 2 -2x + 2 = (x + 1)(x 2 -4x + 2) has roots -1, 2 + √ 2, 2 - √ 2.

You'll notice that the irrational roots, like -1 ± 2i and 2 ± √ 2, are coming up in pairs. In fact, I think precalculus explicitly tells you that the complex roots come in conjugate pairs. More generally, it seems like all the roots of the form a + b √ c come in "conjugate pairs". And you can see why. But a polynomial like x 3 -8x + 4 has no rational roots. (The roots of this are approximately -3.0514, 0.51730, 2.5341.) Or even simpler, x 3 -2 has only one real root, 3 √ 2. These roots, even though they are irrational, have no "conjugate" pairs. Or do they?

Let's try and figure out exactly what's happening. Let α be any complex number. We define a minimal polynomial of α over Q to be a polynomial such that • P (x) has rational coefficients, and leading coefficient 1,

• P (α) = 0.

• The degree of P is as small as possible. We call deg P the degree of α. Note that 100x 2 -200 is also a polynomial of the same degree which has √ 2 as a root; that's why we want to require the polynomial to be monic. That's also why we choose to work in the rational numbers; that way, we can divide by leading coefficients without worrying if we get non-integers.

Why do we care? The point is as follows: suppose we have another polynomial A(x) such that A(α) = 0. Then we claim that P (x) actually divides A(x)! That means that all the other roots of P will also be roots of A.

The proof is by contradiction: if not, by polynomial long division we can find a quotient and remainder Q(x), R(x) such that

A(x) = Q(x)P (x) + R(x)

and R(x) ̸ ≡ 0. Notice that by plugging in x = α, we find that R(α) = 0. But deg R < deg P , and P (x) was supposed to be the minimal polynomial. That's impossible! It follows from this and the monotonicity of the minimal polynomial that it is unique (when it exists), so actually it is better to refer to the minimal polynomial. Definition 47.2.3. Consider an algebraic number α and its minimal polynomial P (which is monic and has rational coefficients). If it turns out the coefficients of P are integers, then we say α is an algebraic integer.

The set of algebraic integers is denoted Z.

Remark 47.2.4 -One can show, using Gauss's Lemma, that if α is the root of any monic polynomial with integer coefficients, then α is an algebraic integer. So in practice, if I want to prove that √ 2 + √ 3 is an algebraic integer, then I only have to say "the polynomial (x 2 -5) 2 -24 works" without checking that it's minimal.

Sometimes for clarity, we refer to elements of Z as rational integers.

Example 47.2.5 (Examples of algebraic integers)

The numbers

4, i = √ -1, 3 √ 2, √ 2 + √ 3

are all algebraic integers, since they are the roots of the monic polynomials x -4, x 2 + 1, x 3 -2 and (x 2 -5) 2 -24.

The number 1 2 has minimal polynomial x -1 2 , so it's an algebraic number but not an algebraic integer. (In fact, the rational root theorem also directly implies that any monic integer polynomial does not have 1 2 as a root!)

There are two properties I want to give for these off the bat, because they'll be used extensively in the tricky (but nice) problems at the end of the section. The first we prove now, since it's very easy: Proposition 47.2.6 (Rational algebraic integers are rational integers) An algebraic integer is rational if and only if it is a rational integer. In symbols,

Z ∩ Q = Z.

Proof. Let α be a rational number. If α is an integer, it is the root of x -α, hence an algebraic integer too.

Conversely, if P is a monic polynomial with integer coefficients such that P (α) = 0 then (by the rational root theorem, say) it follows α must be an integer.

The other is that: Proposition 47.2.7 (Z is a ring and Q is a field)

The algebraic integers Z form a ring. The algebraic numbers Q form a field.

We could prove this now if we wanted to, but the results in the next chapter will more or less do it for us, and so we take this on faith temporarily.

§47.3 Number fields

Prototypical example for this section: Q( √ 2) is a typical number field.

Given any algebraic number α, we're able to consider fields of the form Q(α). Let us write down the more full version.

Definition 47.3.1.

A number field K is a field containing Q as a subfield which is a finite-dimensional Q-vector space. The degree of K is its dimension.

Example 47.3.2 (Prototypical example)

Consider the field

K = Q( √ 2) = a + b √ 2 | a, b ∈ Q .

This is a field extension of Q, and has degree 2 (the basis being 1 and √ 2).

You might be confused that I wrote 

Q( √ 2) (which should permit denominators) instead of Q[ √ 2],

(α) ∼ = Q[α].

Example 47.3.4 (Adjoining an algebraic number)

Let α be the root of some irreducible polynomial P (x) ∈ Q[x]. The field Q(α) is a field extension as well, and the basis is 1, α, α 2 , . . . , α m-1 , where m is the degree of α. In particular, the degree of Q(α) is just the degree of α.

Example 47.3.5 (Non-examples of number fields) R and C are not number fields since there is no finite Q-basis of them. §47.4 Primitive element theorem, and monogenic extensions Prototypical example for this section:

Q( √ 3, √ 5) ∼ = Q( √ 3 + √ 5).

Can you see why?

I'm only putting this theorem here because I was upset that no one told me it was true (it's a very natural conjecture), and I hope to not do the same to the reader. However, I'm not going to use it in anything that follows.

Theorem 47.4.1 (Artin's primitive element theorem)

Every number field K is isomorphic to Q(α) for some algebraic number α.

The proof is left as Problem 53E, since to prove it I need to talk about field extensions first.

The prototypical example

Q( √ 3, √ 5) ∼ = Q( √ 3 + √ 5)

makes it clear why this theorem should not be too surprising. §47.5 A few harder problems to think about Problem 47A. Find a polynomial with integer coefficients which has √ 2 + 3 √ 3 as a root.

Problem 47B (Brazil 2006). Let p be an irreducible polynomial in Q[x] and degree larger than 1. Prove that if p has two roots r and s whose product is 1 then the degree of p is even. The norm of α is defined as

N K/Q (α) := Galois conj of α k .

The trace is defined as

Tr K/Q (α) := k • Galois conj of α .

The exponent of k is a "correction factor" that makes the norm of 5 into 5 2 = 25 when we view 5 as an element of Q(i) rather than an element of Q. For a "generic" element of K, we expect k = 1.

Exercise 48. 1.4 . Use what you know about nested vector spaces to convince yourself that k is actually an integer.

Example 48. 1.5 

(Norm of a + b √ 2) Let α = a + b √ 2 ∈ Q( √ 2) = K.

If b ̸ = 0, then α and K have the degree 2. Thus the only conjugates of α are a ± b √ 2, which gives the norm

(a + b √ 2)(a -b √ 2) = a 2 -2b 2 ,

The trace is (a

-b √ 2) + (a + b √ 2) = 2a.

Nicely, the formula a 2 -2b 2 and 2a also works when b = 0.

Example 48. 1.6 If α is an algebraic integer, its norm and trace are rational integers. 

(Norm of a + b 3 √ 2 + c 3 √ 4) Let α = a + b 3 √ 2 + c 3 √ 4 ∈ Q( √ 3) = K.

(α) = d i=1 σ i (α) and N K/Q (α) = d i=1 σ i (α).

There is exactly d embeddings, regardless of the number of Galois conjugates of α.

Theorem 48. 1 

.10 (Morally correct definition of norm and trace)

Let K be a number field of degree n, and let α ∈ K. Let µ α : K → K denote the map x → αx viewed as a linear map of Q-vector spaces. Then,

• the norm of α equals the determinant det µ α , and

• the trace of α equals the trace Tr µ α .

Since the trace and determinant don't depend on the choice of basis, you can pick whatever basis you want and use whatever definition you got in high school. Fantastic, right?

Example 48. 1.11 (Explicit computation of matrices for a + b √ 2) Consequently, we can interpret µ α as the matrix

Let K = Q( √µ α = a 2b b a .

Of course, the matrix will change if we pick a different basis, but the determinant and trace do not: they are always given by det µ α = a 2 -2b 2 and Tr µ α = 2a.

This interpretation explains why the same formula should work for a + b √ 2 even in the case b = 0.

Proof. I'll prove the result for just the norm; the trace falls out similarly. Set

n = deg α, kn = deg K.

The proof is split into two parts, depending on whether or not k = 1.

Proof if k = 1. Set n = deg α = deg K.

Thus the norm actually is the product of the Galois conjugates. Also, {1, α, . . . , α n-1 } is linearly independent in K, and hence a basis (as dim K = n). Let's use this as the basis for µ α . Let

x n + c n-1 x n-1 + • • • + c 0

be the minimal polynomial of α. Thus µ α (1) = α, µ α (α) = α 2 , and so on, but

µ α (α n-1 ) = -c n-1 α n-1 -• • • -c 0 .

Therefore, µ α is given by the matrix

M =         0 0 0 . . . 0 -c 0 1 0 0 . . . 0 -c 1 0 1 0 . . . 0 -c 2 . . . . . . . . . . . . 0 -c n-2 0 0 0 . . . 1 -c n-1        

.

Thus det M = (-1) n c 0 and we're done by Vieta's formulas. ■

Proof if k > 1. We have nested vector spaces

Q ⊆ Q(α) ⊆ K.

Let e 1 , . . . , e k be a Q(α)-basis for K (meaning: interpret K as a vector space over Q(α), and pick that basis). Since {1, α, . . . , α n-1 } is a Q basis for Q(α), the elements constitute a Q-basis of K. Using this basis, the map µ α looks like

e 1 ,

      M M . . . M       k times

where M is the same matrix as above: we just end up with one copy of our old matrix for each e i . Thus det µ α = (det M ) k , as needed. ■

Question 48.1.12. Verify the result for traces as well.

From this it follows immediately that

N K/Q (αβ) = N K/Q (α) N K/Q (β)

because by definition we have

µ αβ = µ α • µ β ,

and that the determinant is multiplicative. In the same way, the trace is additive.

§48.2 The ring of integers

Prototypical example for this section:

If K = Q( √ 2), then O K = Z[ √ 2]. But if K = Q( √ 5), then O K = Z[ 1+ √ 5 2 ]

. Z makes for better number theory than Q. In the same way, focusing on the algebraic integers of K gives us some really nice structure, and we'll do that here. Definition 48.2.1. Given a number field K, we define

O K := K ∩ Z

to be the ring of integers of K; in other words O K consists of the algebraic integers of K. We do the classical example of a quadratic field now. Before proceeding, I need to write a silly number theory fact.

Exercise 48.2.2 (Annoying but straightforward). Let a and b be rational numbers, and d a squarefree positive integer.

• If d ≡ 2, 3 (mod 4), prove that 2a, a 2 -db 2 ∈ Z if and only if a, b ∈ Z.

• For d ≡ 1 (mod 4), prove that 2a, a 2 -db 2 ∈ Z if and only if a, b ∈ Z OR if a -1 2 , b -1 2 ∈ Z. You'll need to take mod 4. Example 48.2.3 (Ring of integers of K = Q( √ 3))

Let K be as above. We claim that

O K = Z[ √ 3] = m + n √ 3 | m, n ∈ Z .

We set α = a + b √ 3. Then α ∈ O K when the minimal polynomial has integer coefficients.

If b = 0, then the minimal polynomial is x -α = x -a, and thus α works if and only if it's an integer. If b ̸ = 0, then the minimal polynomial is

(x -a) 2 -3b 2 = x 2 -2a • x + (a 2 -3b 2 ).

From the exercise, this occurs exactly for a, b ∈ Z.

Example 48.2.4 (Ring of integers of

K = Q( √ 5))

We claim that in this case

O K = Z 1 + √ 5 2 = m + n • 1 + √ 5 2 | m, n ∈ Z .

The proof is exactly the same, except the exercise tells us instead that for b ̸ = 0, we have both the possibility that a, b ∈ Z or that a, b ∈ Z -1 2 . This reflects the fact that 1+ √ 5 2 is the root of x 2 -x -1 = 0; no such thing is possible with √ 3.

In general, the ring of integers of

K = Q( √ d) is O K =      Z[ √ d] d ≡ 2, 3 (mod 4) Z 1+ √ d 2 d ≡ 1 (mod 4).

What we're going to show is that O K behaves in K a lot like the integers do in Q. First we show K consists of quotients of numbers in O K . In fact, we can do better: Example 48.2.5 (Rationalizing the denominator) For example, consider

K = Q( √ 3). The number x = 1 4+ √

3 is an element of K, but by "rationalizing the denominator" we can write

1 4 + √ 3 = 4 - √ 3 13 .

So we see that in fact, x is 1 13 of an integer in O K .

The theorem holds true more generally.

Theorem 48.2.6

(K = Q • O K )

Let K be a number field, and let x ∈ K be any element. Then there exists an integer n such that nx ∈ O K ; in other words,

x = 1 n α

for some α ∈ O K .

Exercise 48.2.7. Prove this yourself. (Start by using the fact that x has a minimal polynomial with rational coefficients. Alternatively, take the norm.)

Now we are going to show O K is a ring; we'll check it is closed under addition and multiplication. To do so, the easiest route is:

Lemma 48.2.8 (α ∈ Z ⇐⇒ Z[α] finitely generated) Let α ∈ Q.

Then α is an algebraic integer if and only if the abelian group Z[α] is finitely generated.

Proof. Note that α is an algebraic integer if and only if it's the root of some nonzero, monic polynomial with integer coefficients. Suppose first that

α N = c N -1 α N -1 + c N -2 α N -2 + • • • + c 0 .

Then the set 1, α, . 

α N = c 1 b 1 (α) + • • • + c m b m (α).

The above gives us a monic polynomial in α, and the choice of N guarantees it is not zero. So α is an algebraic integer.

Example 48.2.9 ( 1 2 isn't an algebraic integer) We already know 1 2 isn't an algebraic integer. So we expect

Z 1 2 = a 2 m | a, m ∈ Z and m ≥ 0

to not be finitely generated, and this is the case. The set Z is closed under addition and multiplication; i.e. it is a ring. In particular, O K is also a ring for any number field K. In fact, something even better is true. As you saw, for

Q( √ 3) we had O K = Z[ √]; in other words, O K was generated by 1 and √ 3. Something similar was true for Q( √ 5). We claim that in fact, the general picture looks exactly like this.

Theorem 48.2.12 (O K is a free Z-module of rank n)

Let K be a number field of degree n. Then O K is a free Z-module of rank n, i.e. O K ∼ = Z ⊕n as an abelian group. In other words, O K has a Z-basis of n elements as

O K = {c 1 α 1 + • • • + c n-1 α n-1 + c n α n | c i ∈ Z}

where α i are algebraic integers in O K .

(I had a wrong proof here before; the mistake was pointed out 2020-02-12 on https: //math.stackexchange.com/q/3543641/229197. Three years later, I still haven't bothered to fix it, but http://www.jmilne.org/math/CourseNotes/ANT.pdf has a write-up in Proposition 2. 29.) This last theorem shows that in many ways O K is a "lattice" in K. That is, for a number field K we can find α 1 , . . . , α n in O K such that

O K ∼ = α 1 Z ⊕ α 2 Z ⊕ • • • ⊕ α n Z K ∼ = α 1 Q ⊕ α 2 Q ⊕ • • • ⊕ α n Q as abelian groups.

§48.3 On monogenic extensions

Recall that it turned out number fields K could all be expressed as Q(α) for some α. We might hope that something similar is true of the ring of integers: that we can write

O K = Z[θ]

in which case {1, θ, . . . , θ n-1 } serves both as a basis of K and as the Z-basis for O K (here n = [K : Q]). In other words, we hope that the basis of O K is actually a "power basis". This is true for the most common examples we use:

• the quadratic field, and

• the cyclotomic field in Problem 48E † .

Unfortunately, it is not true in general: the first counterexample is

K = Q(α) for α a root of X 3 -X 2 -2X -8.

We call an extension with this nice property monogenic. As we'll later see, monogenic extensions have a really nice factoring algorithm, Theorem 49.5.4.

Remark 48.3.1 (What went wrong with O K ?) -As we have just mentioned above, as an abelian group, O K ∼ = Z 3 , so it's generated by finitely many elements.

In fact, {1, α, β} is a basis of O K , where β = α+α 2 2 . The group generated by {1, α, α 2 } has index 2 in O K -that is, |O K /⟨1, α, α 2 ⟩| = 2, and we misses β.

If we try to pick {1, β, β 2 } as a basis instead, again we get |O K /⟨1, β, β 2 ⟩| = 2, and we misses α. If you explicitly compute it out, you can get β 2 = 3α 2 +7α 2 + 6 = 3β + 2α + 6.

While this is not a proof that the extension is not monogenic, hopefully it gives you a feeling of the structure of O K . §48.4 A few harder problems to think about 

Problem 48A ⋆ . Show that α is a unit of O K (meaning α -1 ∈ O K ) if and only if N K/Q (α) = ±1.

a + b := {a + b | a ∈ a, b ∈ b} a • b := {a 1 b 1 + • • • + a n b n | a i ∈ a, b i ∈ b} .

(Note that infinite sums don't make sense in general rings, which is why in a • b we cut off the sum after some finite number of terms.) You can readily check these are actually ideals. This definition is more natural if you think about it in terms of the generators of a and b. In other words, for sums you append the two sets of generators together, and for products you take products of the generators. Note that for principal ideals, this coincides with "normal" multiplication, for example

(3) • (5) = (15) in Z.

Remark 49.2.3 -Note that for an ideal a and an element c, the set ca = {ca | a ∈ a} is equal to (c) • a. So "scaling" and "multiplying by principal ideals" are the same thing. This is important, since we'll be using the two notions interchangeably.

Remark 49.2.4 -The addition of two ideals does not correspond to the addition of elements -for example, (4) + (6) = (4, 6) = (2), but 4 + 6 = 10. This is the best we can hope for -addition of elements does not make sense for ideals -for example, 1 + 1 = 2 and 1 + (-1) = 0, but as ideals, (1) = (-1).

In fact, addition of ideal is the straightforward generalization of gcd of elementsas you can check in the example above, gcd(4, 6) = 2.

Nevertheless, I hope you agree that a + b is a natural notation, compared to something like (a, b).

Because factorization involves multiplying, instead of adding, the ideals together, we will not need to use the notation a + b any time soon. As mentioned in Remark 5. 3.3 , this also lets us ignore multiplication by units: (-3) = (3).

§49.3 Dedekind domains

Prototypical example for this section: Any O K is a Dedekind domain.

We now define a Dedekind domain as follows.

Definition 49.3.1. An integral domain A is a Dedekind domain if it is Noetherian, integrally closed, and every nonzero prime ideal of A is in fact maximal. (The last condition is the important one.)

Here there's one new word I have to define for you, but we won't make much use of it.

Definition 49.3.2. Let R be an integral domain and let K be its field of fractions. We say R is integrally closed if the only elements a ∈ K which are roots of monic polynomials in R are the elements of R (which are roots of the trivial x -r polynomial).

The interesting condition in the definition of a Dedekind domain is the last one: prime ideals and maximal ideals are the same thing. The other conditions are just technicalities, but "primes are maximal" has real substance.

Example 49.3.3 (Z is a Dedekind domain)

The ring Z is a Dedekind domain. Note that • Z is Noetherian (for obvious reasons).

• Z has field of fractions Q. If f (x) ∈ Z[x] is monic, then by the rational root theorem any rational roots are integers (this is the same as the proof that Z ∩ Q = Z). Hence Z is integrally closed.

• The nonzero prime ideals of Z are (p), which also happen to be maximal.

The case of interest is a ring O K in which we wish to do factorizing. We're now going to show that for any number field K, the ring O K is a Dedekind domain. First, the boring part.

Proposition 49.3.4 (O K integrally closed and Noetherian)

For any number field K, the ring O K is integrally closed and Noetherian.

Proof.

Boring, but here it is anyways for completeness.

Since O K ∼ = Z ⊕n , we get that it's Noetherian. Now we show that O K is integrally closed. Suppose that η ∈ K is the root of some polynomial with coefficients in O K . Thus

η n = α n-1 • η n-1 + α n-2 • η n-2 + • • • + α 0

where α i ∈ O K . We want to show that η ∈ O K as well.

Well, from the above,

O K [η] is finitely generated. . . thus Z[η] ⊆ O K [η] is finitely generated. So η ∈ Z, and hence η ∈ K ∩ Z = O K .

Now let's do the fun part. We'll prove a stronger result, which will re-appear repeatedly.

Theorem 49.3.5 (Important: prime ideals divide rational primes)

Let O K be a ring of integers and p a nonzero prime ideal inside it. Then p contains a rational prime p. Moreover, p is maximal.

For a concrete example, consider (2+i) ⊆ Z[i]. In this case, p = 5, and because p ∈ (2+i), we get that the lattice is periodic in both dimensions with period 5, which implies finitely many points in {a + bi | 0 ≤ a, b ≤ 4, a, b ∈ Z} suffices to cover all cosets modulo the ideal.

The proof of the finiteness of the quotient here is closely related to the statement that the mesh of the lattice is finite, which will be covered in the next section.

Proof. Take any α ̸ = 0 in p. Its Galois conjugates are algebraic integers so their product N(α)/α is in O K (even though each individual conjugate need not be in K). Consequently, N(α) ∈ p, and we conclude p contains some integer.

Then take the smallest positive integer in p, say p. We must have that p is a rational prime, since otherwise p ∋ p = xy implies one of x, y ∈ p. This shows the first part.

We now do something pretty tricky to show p is maximal. Look at O K /p; since p is prime it's supposed to be an integral domain. . . but we claim that it's actually finite! To do this, we forget that we can multiply on O K . Recalling that O K ∼ = Z ⊕n as an abelian group, we obtain a map

F p ⊕n ∼ = O K /(p) ↠ O K /p.

Hence |O K /p| ≤ p n is finite. Since finite integral domains are fields (Problem 5D ⋆ ) we are done.

Since every nonzero prime p is maximal, we now know that O K is a Dedekind domain. Note that this tricky proof is essentially inspired by the solution to Problem 5G. • An integral domain admits unique factorization of elements if the prime elements and the irreducible elements are the same.

• An integral domain admits unique factorization of ideals if the prime ideals and the maximal ideals are the same.

Notice the similarity -in either case, the Noetherian condition is "merely" to ensure that, if you keep extracting prime factors, you will terminate in a finite time.

Example 49.4.2 (What went wrong if A is not integrally closed?)

Consider A = 2Z, which is an ideal of Z. Clearly, every nonzero prime ideal is maximal. Nevertheless, in A, (2 60) is not a prime ideal (so of course it isn't a maximal ideal), but we cannot break it down into, for example, (2 • 3) • (5).

• 3 • 5) = (

Theorem 49.4.3 (Prime factorization works)

Let a be a nonzero proper ideal of a Dedekind domain A. Then a can be written as a finite product of nonzero prime ideals p i , say

a = p e 1

1 p e 2 2 . . . p eg g and this factorization is unique up to the order of the p i . Moreover, a divides b if and only if for every prime ideal p, the exponent of p in a is less than the corresponding exponent in b.

I won't write out the proof, but I'll describe the basic method of attack. Section 3 of [Ul08] does a nice job of explaining it. When we proved the fundamental theorem of arithmetic, the basic plot was:

(1) Show that if p is a rational prime 1 (2) Use strong induction to show that every N > 1 can be written as the product of primes (easy).

(3) Show that if p 1 . . . p m = q 1 . . . q n for some primes (not necessarily unique), then p 1 = q i for some i, say q 1 .

(4) Divide both sides by p 1 and use induction.

1 Note that the kindergarten definition of a prime is that "p isn't the product of two smaller integers". This isn't the correct definition of a prime: the definition of a prime is that p | bc means p | b or p | c. The kindergarten definition is something called "irreducible". Fortunately, in Z, primes and irreducibles are the same thing, so no one ever told you that your definition of "prime" was wrong.

What happens if we try to repeat the proof here? We get step 1 for free, because we're using a better definition of "prime". We can also do step 3, since it follows from step 1 Well, it turns out that we can define the inverse a -1 of an ideal, and I'll do so by the end of this chapter. You then need to check that a • a -1 = (1) = A. In fact, even this isn't easy. You have to check it's true for prime ideals p, then prove prime factorization, and then prove that this is true. Moreover, a -1 is not actually an ideal, so you need to work in the field of fractions K instead of A.

So the main steps in the new situation are as follows:

(1) First, show that every ideal a divides p 1 . . . p g for some finite collection of primes.

(This is an application of Zorn's Lemma.)

(2) Define p -1 and show that pp -1 = (1).

(3) Show that a factorization exists (again using Zorn's Lemma).

(4) Show that it's unique, using the new inverse we've defined.

Finally, let me comment on how nice this is if A is a PID (like Z). Thus every element a ∈ A is in direct correspondence with an ideal (a). Now suppose (a) factors as a product of ideals p i = (p i ), say, (a) = (p 1 ) e 

= Q( √ d), then O K =    Z[ √ d] d ≡ 2, 3 (mod 4) Z 1+ √ d 2 d ≡ 1 (mod 4).

Also, recall that the norm of a + b √ -d is given by a 2 + db 2 .

Example 49.5.1 (Factoring 6 in the integers of Q( √ -5))

Let O K = Z[ √ -5] arise from K = Q( √ -5).

We've already seen that

(6) = (2) • (3) = 1 + √ -5 1 - √ -5

and you can't get any further with these principal ideals. But let

p = 1 + √ -5, 2 = 1 - √ -5, 2 and q 1 = (1 + √ -5, 3), q 2 = (1 - √ -5, 3).

Then it turns out (6) = p 2 q 1 q 2 . More specifically, (2) = p 2 , (3) = q 1 q 2 , and (1 + √ -5) = pq 1 and (1 -√ -5) = pq 2 . (Proof in just a moment.)

I want to stress that all our ideals are computed relative to O K . So for example,

(2) = {2x | x ∈ O K } .

How do we know in this example that p is prime/maximal? (Again, these are the same since we're in a Dedekind domain.) Answer: look at O K /p and see if it's a field. There is a trick to this: we can express

O K = Z[ √ -5] ∼ = Z[x]/(x 2 + 5).

So when we take that mod p, we get that

O K /p = Z[x]/(x 2 + 5, 2, 1 + x) ∼ = F 2 [x]/(x 2 + 5, x + 1)

as rings.

Question 49.5.2. Conclude that O K /p ∼ = F 2 , and satisfy yourself that q 1 and q 2 are also maximal.

I should give an explicit example of an ideal multiplication: let's compute

q 1 q 2 = (1 + √ -5)(1 - √ -5), 3(1 + √ -5), 3(1 - √ -5), 9 = 6, 3 + 3 √ -5, 3 -3 √ -5, 9 = 6, 3 + 3 √ -5, 3 -3 √ -5, 3 = (3)

where we first did 9 -6 = 3 (think Euclidean algorithm!), then noted that all the other generators don't contribute anything we don't already have with the 3 (again these are ideals computed in O K ). You can do the computation for p 2 , pq 1 , pq 2 in the same way.

Finally, it's worth pointing out that we should quickly verify that p ̸ = (x) for some x; in other words, that p is not principal. Assume for contradiction that it is. Then x divides both 1 + √ -5 and 2, in the sense that 1 + √ -5 = α 1 x and 2 = α 2 x for some α 1 , α 2 ∈ O K . (Principal ideals are exactly the "multiples" of x, so (x) = xO K .) Taking the norms, we find that N K/Q (x) divides both

N K/Q (1 + √ -5) = 6 and N K/Q (2) = 4.

Since p ̸ = (1), x cannot be a unit, so its norm must be 2. But there are no elements of norm 2 = a 2 + 5b 2 in O K .

Example 49.5.3 (Factoring 3 in the integers of Q( √ -17))

Let O K = Z[ √ -17] arise from K = Q( √ -17). We know O K ∼ = Z[x]/(x 2 + 17). Now O K /3O K ∼ = Z[x]/(3, x 2 + 17) ∼ = F 3 [x]/(x 2 -1).

This already shows that (3) cannot be a prime (i.e. maximal) ideal, since otherwise our result should be a field. Anyways, we have a projection

O K ↠ F 3 [x]/ ((x -1)(x + 1)) .

Let q 1 be the pre-image of (x -1) in the image, that is,

q 1 = (3, √ -17 -1).

Similarly, q 2 = (3, √ -17 + 1).

We have O K /q 1 ∼ = F 3 , so q 1 is maximal (prime). Similarly q 2 is prime. Magically, you can check explicitly that q 1 q 2 = (3).

Hence this is the factorization of (3) into prime ideals.

The fact that q 1 q 2 = (3) looks magical, but it's really true:

q 1 q 2 = (3, √ -17 -1)(3, √ -17 + 1) = (9, 3 √ -17 + 3, 3 √ -17 -3, 18) = (9, 3 √ -17 + 3, 6) = (3, 3 √ -17 + 3, 6) = (3).

In fact, it turns out this always works in general: given a rational prime p, there is an algorithm to factor p in any O K of the form Z[θ].

Theorem 49.5.4 (Factoring algorithm / Dedekind-Kummer theorem)

Let K be a number field. Let θ ∈ O K with |O K /Z[θ]| = j < ∞, and let p be a prime not dividing j. Then (p) = pO K is factored as follows:

Let f be the minimal polynomial of θ and factor f mod p as

f ≡ g i=1 (f i ) e i (mod p).

Then p i = (f i (θ), p) is prime for each i and the factorization of (p) is

O K ⊇ (p) = g i=1 p e i i .

In particular, if K is monogenic with O K = Z[θ] then j = 1 and the theorem applies for all primes p.

In almost all our applications in this book, K will be monogenic; i.e. j = 1. Here ψ denotes the image in

F p [x] of a polynomial ψ ∈ Z[x].

Question 49.5.5. There are many possible pre-images f i we could have chosen (for example if f i = x 2 + 1 (mod 3), we could pick f i = x 2 + 3x + 7.) Why does this not affect the value of p i ?

Note that earlier, we could check the factorization worked for any particular case. The proof that this works is much the same, but we need one extra tool, the ideal norm. After that we leave the proposition as Problem 49E. We now have a neat theory of factoring ideals of A, just like factoring the integers. Now note that our factorization of Z naturally gives a way to factor elements of Q; just factor the numerator and denominator separately.

This algorithm gives us a concrete way to compute prime factorizations of (p) in any monogenic number field with O

K = Z[θ].

Let's make the analogy clearer. The analogue of a rational number is as follows.

Definition 49.6.1. Let A be a Dedekind domain with field of fractions K. A fractional ideal J of K is a set of the form

J = 1 x

• a where x ∈ A, and a is an integral ideal.

For emphasis, ideals of A will be sometimes referred to as integral ideals.

You might be a little surprised by this definition: one would expect that a fractional ideal should be of the form a b for some integral ideals a, b. But in fact, it suffices to just take x ∈ A in the denominator. The analogy is that when we looked at O K , we found that we only needed integer denominators: 1 4-√ 3 = 1 13 (4 + √ 3). Similarly here, it will turn out that we only need to look at 1 x • a rather than a b , and so we define it this way from the beginning. See Problem 49D † for a different equivalent definition.

Example 49.6.2

( 5 2 Z is a fractional ideal) The set 5 2 Z = 5 2 n | n ∈ Z = 1 2 (5)

is a fractional ideal of Z.

Now, as we prescribed, the fractional ideals form a multiplicative group:

Theorem 49.6.3 (Fractional ideals form a group)

Let A be a Dedekind domain and K its field of fractions. For any integral ideal a, the set

a -1 = {x ∈ K | xa ⊆ (1) = A}

is a fractional ideal with aa -1 = (1).

(This result is nontrivial. To prove aa -1 = (1), one approach is to prove it first for prime a, then consider the factorization of a into prime ideals.) Definition 49.6.4. Thus nonzero fractional ideals of K form a group under multiplication with identity (1) = A. This ideal group is denoted J K .

Example 49.6.5 ((3) -1 in Z)

Please check that in Z we have

(3) -1 = 1 3 n | n ∈ Z = 1 3 Z.

It follows that every fractional ideal J can be uniquely written as

J = i p n i i • q -m i i

where n i and m i are positive integers. In fact, a is an integral ideal if and only if all its exponents are nonnegative, just like the case with integers. So, a perhaps better way to think about fractional ideals is as products of prime ideals, possibly with negative exponents.

§49.7 The ideal norm

One last tool is the ideal norm, which gives us a notion of the "size" of an ideal. Example 49.7.2 (Ideal norm of (5) in the Gaussian integers)

Let K = Q(i), O K = Z[i].

Consider the ideal (5) in O K . We have that

O K /(5) ∼ = {a + bi | a, b ∈ Z/5Z}

so (5) has ideal norm 25, corresponding to the fact that O K /(5) has 5 2 = 25 elements.

Example 49.7.3 (Ideal norm of (2 + i) in the Gaussian integers)

You'll notice that

O K /(2 + i) ∼ = F 5

since mod 2 + i we have both 5 ≡ 0 and i ≡ -2. (Indeed, since (2 + i) is prime we had better get a field!) Thus N ((2 + i)) = 5; similarly N ((2 -i)) = 5.

Thus the ideal norm measures how "roomy" the ideal is: that is, (5) is a lot more spaced out in Z[i] than it is in Z. (This intuition will be important when we will actually view O K as a lattice.) I unfortunately won't prove these properties, though we already did (a) in our proof that O K was a Dedekind domain.

The fact that N is completely multiplicative lets us also consider the norm of a fractional ideal J by the natural extension

J = i p n i i • i q -m i i =⇒ N(J) := i N(p i ) n i i N(q i ) m i .

Thus N is a natural group homomorphism Problem 49C (Fermat's little theorem). Let p be a prime ideal in some ring of integers O K . Show that for α ∈ O K , α N(p) ≡ α (mod p).

J K → Q \ {0}.

Problem 49D † .

Let A be a Dedekind domain with field of fractions K, and pick J ⊆ K. Show that J is a fractional ideal if and only if (i) J is closed under addition and multiplication by elements of A, and

(ii) J is finitely generated as an abelian group.

More succinctly: J is a fractional ideal ⇐⇒ J is a finitely generated A-module.

Problem 49E. In the notation of Theorem 49.5.4, let I = g i=1 p e i i . Assume for simplicity that K is monogenic, hence 

O K = Z[θ].

50 Minkowski bound and class groups

We now have a neat theory of unique factorization of ideals. In the case of a PID, this in fact gives us a UFD. Sweet.

We'll define, in a moment, something called the class group which measures how far O K is from being a PID; the bigger the class group, the farther O K is from being a PID. In particular, O K is a PID if it has trivial class group.

Then we will provide some inequalities which let us put restrictions on the class group; for instance, this will let us show in some cases that the class group must be trivial. Astonishingly, the proof will use Minkowski's theorem, a result from geometry. §50.1 The class group Prototypical example for this section: PID's have trivial class group.

Let K be a number field, and let J K denote the multiplicative group of fractional ideals of O K . Let P K denote the multiplicative group of principal fractional ideals: those of the form (x) = xO K for some x ∈ K.

Question 50.1.1. Check that P K is also a multiplicative group. (This is really easy: name

xO K • yO K and (xO K ) -1 .)

As J K is abelian, we can now define the class group (or ideal class group) to be the quotient Cl K := J K /P K .

The elements of Cl K are called classes.

Equivalently,

The class group Cl K is the set of nonzero fractional ideals modulo scaling by a constant in K.

You can also think of the classes as the "shapes" of the ideals, as two ideals belong to the same class if and only if they're isomorphic as O K -modules.

In particular, Cl K is trivial if all ideals are principal, since the nonzero principal ideals are the same up to scaling.

The size of the class group is called the class number. It's a beautiful theorem that the class number is always finite, and the bulk of this chapter will build up to this result.

It requires several ingredients. §50.2 The discriminant of a number field

Prototypical example for this section: Quadratic fields.

Let's say I have

K = Q( √ 2).

As we've seen before, this means

O K = Z[ √ 2], meaning O K = a + b √ 2 | a, b ∈ Z .

The key insight now is that you might think of this as a lattice: geometrically, we want to think about this the same way we think about Z 2 . Perversely, we might try to embed this into Q 2 by sending a + b √ 2 to (a, b). But this is a little stupid, since we're rudely making K, which somehow lives inside R and is "one-dimensional" in that sense, into a two-dimensional space. It also depends on a choice of basis, which we don't like. A better way is to think about the fact that there are two embeddings σ 1 : K → C and σ 2 : K → C, namely the identity, and conjugation:

σ 1 (a + b √ 2) = a + b √ 2 σ 2 (a + b √ 2) = a -b √ 2.

Fortunately for us, these embeddings both have real image. This leads us to consider the set of points (σ

1 (α), σ 2 (α)) ∈ R 2 as α ∈ K.

This lets us visualize what O K looks like in R 2 . The points of K are dense in R 2 , but the points of O K cut out a lattice.

-2

-1 0 1 2 - √ 2 √ 2 1 + √ 2

To see how big the lattice is, we look at how {1, √ 2}, the generators of O K , behave. The point corresponding to a + b √ 2 in the lattice is

a • (1, 1) + b • ( √ 2, - √ 2).

The mesh of the lattice1 is defined as the hypervolume of the "fundamental parallelepiped" I've colored blue above. For this particular case, it ought to be equal to the area of that parallelogram, which is

det 1 - √ 2 1 √ 2 = 2 √ 2.

The definition of the discriminant is precisely this, except with an extra square factor (since permutation of rows could lead to changes in sign in the matrix above). Problem 51B ⋆ shows that the squaring makes ∆ K an integer.

To make the next definition, we invoke:

Theorem 50.2.1 (The n embeddings of a number field)

Let K be a number field of degree n. Then there are exactly n field homomorphisms K → C, say σ 1 , . . . , σ n , which fix Q.

Proof. Deferred to Theorem 53.3.1, once we have the tools of Galois theory.

In fact, in Theorem 53. 3.4 we see that for α ∈ K, we have that σ i (α) runs over the conjugates of α as i = 1, . . . , n. It follows that

Tr K/Q (α) = n i=1 σ i (α) and N K/Q (α) = n i=1 σ i (α).

This allows us to define:

Definition 50.2.2. Suppose α 1 , . . . , α n is a Z-basis of O K .

The discriminant of the number field K is defined by

∆ K := det    σ 1 (α 1 ) . . . σ n (α 1 ) . . . . . . . . . σ 1 (α n ) . . . σ n (α n )    2 .

This does not depend on the choice of the {α i }; we will not prove this here.

Example 50.2.3

(Discriminant of K = Q( √ 2))

We have

O K = Z[ √ 2

] and as discussed above the discriminant is

∆ K = (-2 √ 2) 2 = 8.

Example 50.2.4 (Discriminant of Q(i))

Let K = Q(i). We have O K = Z[i] = Z ⊕ iZ.

The embeddings are the identity and complex conjugation which take 1 to (1, 1) and i to (i, -i). So

∆ K = det 1 1 i -i 2 = (-2i) 2 = -4.

This example illustrates that the discriminant need not be positive for number fields which wander into the complex plane (the lattice picture is a less perfect analogy).

But again, as we'll prove in the problems the discriminant is always an integer.

Example 50.2.5 (Discriminant of Q( √ 5))

Let K = Q( √ 5). This time, O K = Z ⊕ 1+ √ 5

2 Z, and so the discriminant is going to look a little bit different. The embeddings are still a

+ b √ 5 → a + b √ 5, a -b √ 5.

Applying this to the Z-basis 1, 1+ √ 5 2 , we get

∆ K = det 1 1 1+ √ 5 2 1- √ 5 2 2 = (- √ 5) 2 = 5.

Exercise 50.2.6. Extend all this to show that if

K = Q( √ d) for d ̸ = 1 squarefree, we have ∆ K = d if d ≡ 1 (mod 4) 4d if d ≡ 2, 3 (mod 4).

Actually, let me point out something curious: recall that the polynomial discriminant of Ax 2 + Bx + C is B 2 -4AC. Then:

• In the d ≡ 1 (mod 4) case, ∆ K is the discriminant of x 2 -x -d-1 4 , which is the minimal polynomial of 1 2 (1 + √ d). Of course, O K = Z[ 1 2 (1 + √ d)]. • In the d ≡ 2, 3 (mod 4) case, ∆ K is the discriminant of x 2 -d which is the minimal polynomial of √ d. Once again, O K = Z[ √ d].

This is not a coincidence! Problem 51C ⋆ asserts that this is true in general; hence the name "discriminant". §50.3 The signature of a number field Prototypical example for this section: Q( 100 √ 2) has signature (2, 49).

In the example of K = Q(i), we more or less embedded K into the space C. However, K is a degree two extension, so what we'd really like to do is embed it into R 2 . To do so, we're going to take advantage of complex conjugation.

Let K be a number field and σ 1 , . . . , σ n be its embeddings. We distinguish between the real embeddings (which map all of K into R) and the complex embeddings (which map some part of K outside R). Notice that if σ is a complex embedding, then so is the conjugate σ ̸ = σ; hence complex embeddings come in pairs. Definition 50.3.1. Let K be a number field of degree n, and set r 1 = number of real embeddings r 2 = number of pairs of complex embeddings.

The signature of K is the pair (r 1 , r 2 ). Observe that r 1 + 2r 2 = n. (c) Q(i) has signature (0, 1).

(d) Let K = Q( 3 √ 2)

, and let ω be a cube root of unity. The elements of K are

K = a + b 3 √ 2 + c 3 √ 4 | a, b, c ∈ Q .

Then the signature is (1, 1), because the three embeddings are

σ 1 : 3 √ 2 → 3 √ 2, σ 2 : 3 √ 2 → 3 √ 2ω, σ 3 : 3 √ 2 → 3 √ 2ω 2 .

The first of these is real and the latter two are conjugate pairs.

Example 50.3.3 (Even more signatures)

In the same vein Q( 99 √ 2) and Q( 100 √ 2) have signatures (1, 49) and (2, 49).

Question 50.3.4. Verify the signatures of the above two number fields.

From now on, we will number the embeddings of K in such a way that σ 1 , σ 2 , . . . , σ r 1 are the real embeddings, while

σ r 1 +1 = σ r 1 +r 2 +1 , σ r 1 +2 = σ r 1 +r 2 +2 , . . . , σ r 1 +r 2 = σ r 1 +2r 2 .

are the r 2 pairs of complex embeddings. We define the canonical embedding of K as

K ι → R r 1 × C r 2 by α → (σ 1 (α), . . . , σ r 1 (α), σ r 1 +1 (α), . . . , σ r 1 +r 2 (α)) .

All we've done is omit, for the complex case, the second of the embeddings in each conjugate pair. This is no big deal, since they are just conjugates; the above tuple is all the information we need.

For reasons that will become obvious in a moment, I'll let τ denote the isomorphism

τ : R r 1 × C r 2 ∼ -→ R r 1 +2r 2 = R n

by breaking each complex number into its real and imaginary part, as

α → σ 1 (α), . . . , σ r 1 (α), Re σ r 1 +1 (α), Im σ r 1 +1 (α), Re σ r 1 +2 (α), Im σ r 1 +2 (α), . . . , Re σ r 1 +r 2 (α), Im σ r 1 +r 2 (α) .

Example 50.

3.5 (Example of canonical embedding)

As before let K = Q( 3 √ 2) and set

σ 1 : 3 √ 2 → 3 √ 2, σ 2 : 3 √ 2 → 3 √ 2ω, σ 3 : 3 √ 2 → 3 √ 2ω 2

where ω = -1 2 + √ 3 2 i, noting that we've already arranged indices so σ 1 = id is real while σ 2 and σ 3 are a conjugate pair. So the embeddings

K ι → R × C ∼ -→ R 3 are given by α ι -→ (σ 1 (α), σ 2 (α)) τ -→ (σ 1 (α), Re σ 2 (α), Im σ 2 (α)) .

For concreteness, taking α = 9 + 3 √ 2 gives

9 + 3 √ 2 ι → 9 + 3 √ 2, 9 + 3 √ 2ω = 9 + 3 √ 2, 9 - 1 2 3 √ 2 + 6 √ 108 2 i ∈ R × C τ → 9 + 3 √ 2, 9 - 1 2 3 √ 2, 6 √ 108 2 ∈ R 3 .

Now, the whole point of this is that we want to consider the resulting lattice when we take O K . In fact, we have:

Lemma 50.3.6

Consider the composition of the embeddings

K → R r 1 × C r 2 ∼ -→ R n . Then as before, O K becomes a lattice L in R n , with mesh equal to 1 2 r 2 |∆ K |.

Proof. Fun linear algebra problem (you just need to manipulate determinants). Left as Problem 50D.

From this we can deduce: Lemma 50.3.7

Consider the composition of the embeddings

K → R r 1 × C r 2 ∼ -→ R n .

Let a be an ideal in O K . Then the image of a is a lattice L a in R n with mesh equal to

N(a) 2 r 2 |∆ K |. Sketch of Proof. Let d = N(a) := |O K /a|.

Then in the lattice L a , we somehow only take 1 d th of the points which appear in the lattice L, which is why the area increases by a factor of N(a). To make this all precise I would need to do a lot more with lattices and geometry than I have space for in this chapter, so I will omit the details. But I hope you can see why this is intuitively true.

§50.4 Minkowski's theorem

Now I can tell you why I insisted we move from R r 1 × C r 2 to R n . In geometry, there's a really cool theorem of Minkowski's that goes as follows. Sketch of Proof. Part (a) is surprisingly simple and has a very olympiad-esque solution: it's basically Pigeonhole on areas. We'll prove part (a) in the special case n = 2, L = Z 2 for simplicity as the proof can easily be generalized to any lattice and any n. Thus we want to show that any such convex set S with area more than 4 contains a lattice point.

Dissect the plane into 2 × 2 squares

[2a -1, 2a + 1] × [2b -1, 2b + 1]

and overlay all these squares on top of each other. By the Pigeonhole Principle, we find there exist two points p ̸ = q ∈ S which is mapped to the same point. Since S is symmetric, -q ∈ S. Then 1 2 (p -q) ∈ S (convexity) and is a nonzero lattice point. I'll briefly sketch part (b): the idea is to consider (1+ε)S for ε > 0 (this is "S magnified by a small factor 1 + ε"). This satisfies condition (a). So for each ε > 0 the set of nonzero lattice points in (1 + ε)S, say S ε , is a finite nonempty set of (discrete) points (the "finite" part follows from the fact that (1 + ε)S is bounded). So there has to be some point that's in S ε for every ε > 0 (why?), which implies it's in S.

§50.5 The trap box

The last ingredient we need is a set to apply Minkowski's theorem to. I propose: Definition 50.5.1. Let M be a positive real. In R r 1 × C r 2 , define the box S to be the set of points (x 1 , . . . , x r 1 , z 1 , . . . , z r 2 ) such that

r 1 i=1 |x i | + 2 r 2 j=1 |z j | ≤ M.

Note that this depends on the value of M .

Think of this box as a mousetrap: anything that falls in it is going to have a small norm, and our goal is to use Minkowski to lure some nonzero element into it.

S

-2 -1 0 1 2 - √ 2 √ 2 1 + √ 2

That is, suppose α ∈ a falls into the box I've defined above, which means

M ≥ r 1 i=1 |σ i (α)| + 2 r 1 +r 2 i=r 1 +1 |σ i (α)| = n i=1 |σ i (α)| ,

where we are remembering that the last few σ's come in conjugate pairs. This looks like the trace, but the absolute values are in the way. So instead, we apply AM-GM to obtain:

Lemma 50.5.2 (Effect of the mousetrap)

Let α ∈ O K , and suppose ι(α) is in S (where ι :

K → R r 1 × C r 2 as usual). Then N K/Q (α) = n i=1 |σ i (α)| ≤ M n n .

The last step we need to do is compute the volume of the box. This is again some geometry I won't do, but take my word for it:

Lemma 50.5.3 (Size of the mousetrap)

Let τ : R r 1 × C r 2 ∼ -→ R n as before. Then the image of S under τ is a convex, compact, centrally symmetric set with volume

2 r 1 • π 2 r 2 • M n n! .

Question 50.5.4. (Sanity check) Verify that the above is correct for the signatures (r 1 , r 2 ) = (2, 0) and (r 1 , r 2 ) = (0, 1), which are the possible signatures when n = 2.

§50.6 The Minkowski bound

We can now put everything we have together to obtain the great Minkowski bound.

Theorem 50.6.1 (Minkowski bound)

Let a ⊆ O K be any nonzero ideal. Then there exists 0 ̸ = α ∈ a such that

N K/Q (α) ≤ 4 π r 2 n! n n |∆ K | • N(a).

Proof. This is a matter of putting all our ingredients together. Let's see what things we've defined already:

K R r 1 × C r 2 R n box S τ img (S) with volume 2 r 1 π 2 r 2 M n n! O K Lattice L with mesh 2 -r 2 |∆ K | a Lattice L a with mesh 2 -r 2 |∆ K | N(a) ⊃ ι τ

Pick a value of M such that the mesh of L a equals 2 -n of the volume of the box. Then Minkowski's theorem gives that some 0 ̸ = α ∈ a lands inside the box -the mousetrap is configured to force

N K/Q (α) ≤ 1 n n M n . The correct choice of M is M n = M n • 2 n • mesh vol box = 2 n • n! 2 r 1 • π 2 r 2 • 2 -r 2 |∆ K | N(a)

which gives the bound after some arithmetic. 

(α) • a -1 ≤ M K

where M K is some constant depending on K, and α ∈ a.

Question 50.7.2. Show that (α) • a -1 is an integral ideal. (Unwind definitions.)

But in the class group we mod out by principal ideals like (α). If we shut our eyes for a moment and mod out, the above statement becomes "N(a -1 ) ≤ M K ". The precise statement of this is Corollary 50.7.3

Let K be a number field, and pick a fractional ideal J. Then we can find α such that b = (α) • J is integral and N(b) ≤ M K .

Proof. For fractional ideals I and J write I ∼ J to mean that I = (α)J for some α; then Cl K is just modding out by ∼. Let J be a fractional ideal. Then J -1 is some other fractional ideal. By definition, for some α ∈ O K we have that αJ -1 is an integral ideal a. The Minkowski bound tells us that for some x ∈ a, we have N(xa -1 ) ≤ M K . But xa -1 ∼ a -1 = (αJ -1 ) -1 ∼ J.

Corollary 50.7.4 (Finiteness of class group)

Class groups are always finite.

Proof. For every class in Cl K , we can identify an integral ideal a with norm less than M K . We just have to show there are finitely many such integral ideals; this will mean there are finitely many classes. Suppose we want to build such an ideal a = p e 1 1 . . . p em m . Recall that a prime ideal p i must have some rational prime p inside it, meaning p i divides (p) and p divides N(p i ). So let's group all the p i we want to build a with based on which (p) they came from.

(2)

(3) (5) . . . By computing the actual value of M K , we can quite literally build the entire "cherry tree" mentioned in the previous proof. Let's give an example how! Proposition 50.8.3

The field Q( √ -67) has class number 1.

Proof. Since K = Q( √ -67) has signature (0, 1) and discriminant ∆ K = -67 (since -67 ≡ 1 (mod 4)) we can compute

M K = 4 π 1 • 2! 2 2 √ 67 ≈ 5.2.

That means we can cut off the cherry tree after (2), (3), (5), since any cherries on these branches will necessarily have norm ≥ M K . We now want to factor each of these in

O K = Z[θ], where θ = 1+ √ -67 2

has minimal polynomial x 2 -x + 17. But something miraculous happens:

• When we try to reduce x 2 -x + 17 (mod 2), we get an irreducible polynomial

x 2 -x + 1. By the factoring algorithm (Theorem 49.5.4) this means (2) is prime.

• Similarly, reducing mod 3 gives x 2 -x + 2, which is irreducible. This means (3) is prime.

• Finally, for the same reason, (5) is prime.

It's our lucky day; all of the ideals (2), (3), (5) are prime (already principal). To put it another way, each of the three branches has only one (large) cherry on it. That means any time we put together an integral ideal with norm ≤ M K , it is actually principal. In fact, these guys have norm 4, 9, 25 respectively. . . so we can't even touch (3) and ( 5), and the only ideals we can get are ( 1) and (2) (with norms 1 and 4). Now we claim that's all. Pick a fractional ideal J. By Corollary 50.7.3, we can find an integral ideal b ∼ J with N(b) ≤ M K . But by the above, either b = (1) or b = (2), both of which are principal, and hence trivial in Cl K . So J is trivial in Cl K too, as needed.

Let's do a couple more.

Theorem 50.8.4 (Gaussian integers Z[i] form a UFD)

The field Q(i) has class number 1.

Proof. This is O K where K = Q(i), so we just want Cl K to be trivial. We have

M K = 2 π √ 4 < 2.

So every class has an integral ideal of norm b satisfying

N(b) ≤ 4 π 1 • 2! 2 2 • √ 4 = 4 π < 2.

Well, that's silly: we don't have any branches to pick from at all. In other words, we can only have b = (1).

Here's another example of something that still turns out to be unique factorization, but this time our cherry tree will actually have cherries that can be picked.

Proposition 50.8.5 (Z[ √ 7] is a UFD)

The field Q( √ 7) has class number 1.

Proof. First we compute the Minkowski bound.

Question 50.8.6. Check that M K ≈ 2.646.

So this time, the only branch is (2). Let's factor (2) as usual: the polynomial x 2 + 7 reduces as (x -1)(x + 1) (mod 2), and hence

(2) = 2, √ 7 -1 2, √ 7 + 1 .

Oops! We now have two cherries, and they both seem reasonable. But actually, I claim that 2, √ 7 -1 = 3 -√ 7 .

However, you can compute p2 = (2) so p 2 and (1) are in the same class group; that is, they are trivial. In particular, the class group has order at most 4. From now on, let [a] denote the class (member of the class group) that a is in. Since p isn't principal (so [p] ̸ = [(1)]), it follows that p has order two. So Lagrange's theorem says that Cl K has order either 2 or 4. Now we claim [q 1 ] 2 ̸ = [(1)], which implies that q 1 has order greater than 2. If not, q 2 1 is principal. We know N(q 1 ) = 3, so this can only occur if q 2 1 = (3); this would force q 1 = q 2 . This is impossible since q 1 + q 2 = (1).

Thus, q 1 has even order greater than 2. So it has to have order 4. From this we deduce 

Cl K ∼ = Z/4Z

→ R r 1 × C r 2 ∼ -→ R n . Show that the image of O K ⊆ K has mesh equal to 1 2 r 2 |∆ K |. Problem 50E.

Let p ≡ 1 (mod 4) be a prime. Show that there are unique integers a > b > 0 such that a 2 + b 2 = p.

Problem 50F (Korea national olympiad 2014). Let p be an odd prime and k a positive integer such that p | k 2 + 5. Prove that there exist positive integers m, n such that p 2 = m 2 + 5n 2 . I'll remind you that the discriminant of a number field K is given by

∆ K := det    σ 1 (α 1 ) . . . σ n (α 1 ) . . . . . . . . . σ 1 (α n ) . . . σ n (α n )    2

where α 1 , . 

∆ K = (-1)

p-1

2 p p-2 .

Problem 51B ⋆ (Trace representation of ∆ K ). Let α 1 , . . . , α n be a basis for O K . Prove that

∆ K = det       Tr K/Q (α 2 1 ) Tr K/Q (α 1 α 2 ) . . . Tr K/Q (α 1 α n ) Tr K/Q (α 2 α 1 ) Tr K/Q (α 2 2 ) . . . Tr K/Q (α 2 α n ) . . . . . . . . . . . . Tr K/Q (α n α 1 ) Tr K/Q (α n α 2 ) . . . Tr K/Q (α n α n )       .

In particular, ∆ K is an integer.

Problem 51C ⋆ (Root representation of ∆ K ). The discriminant of a quadratic polynomial Ax 2 + Bx + C is defined as B 2 -4AC. More generally, the polynomial discriminant of a polynomial

f ∈ Z[x] of degree n is ∆(f ) := c 2n-2 1≤i<j≤n (z i -z j ) 2

where z 1 , . . . , z n are the roots of f , and c is the leading coefficient of f . Suppose K is monogenic with

O K = Z[θ]. Let f denote the minimal polynomial of θ (hence monic). Show that ∆ K = ∆(f ). Problem 51D. Show that if K ̸ = Q is a number field then |∆ K | > 1.

Problem 51E (Brill's theorem). For a number field K with signature (r 1 , r 2 ), show that ∆ K > 0 if and only if r 2 is even.

Problem 51F (Stickelberger theorem). Let K be a number field. Prove that ∆ K ≡ 0 or 1 (mod 4).

52 Bonus: Let's solve Pell's equation! This is an optional aside, and can be safely ignored. (On the other hand, it's pretty short.) §52.1 Units Prototypical example for this section: ±1, roots of unity, 3 -2 √ 2 and its powers.

Recall according to Problem 48A ⋆ that α ∈ O K is invertible if and only if

N K/Q (α) = ±1.

We let O × K denote the set of units of O K . 

O K = Z[ √ 3] (from K = Q( √)) then the number 2 + √ 3 is a unit, as its norm is

N K/Q (2 + √ 3) = 2 2 -3 • 1 2 = 1.

Alternatively, just note that the inverse 2 -√ 3 ∈ O K as well:

2 - √ 3 2 + √ 3 = 1.

Either way, 2 -√ 3 is a unit.

4. Given any unit u ∈ O ×

K , all its powers are also units. So for example, (3-2 

√ 2) n is always a unit of Z[ √ 2],

(O K )) (a) If K = Q(i), then O K = Z[i]. So µ(O K ) = {±1, ±i} where K = Q(i). (b) If K = Q( √ 3), then O K = Z[ √ 3]. So µ(O K ) = {±1} where K = Q( √ 3). (c) If K = Q( √ -3), then O K = Z[ 1 2 (1 + √ -3)]. So µ(O K ) = ±1, ±1 ± √ -3 2 where K = Q( √ -3)

where the ±'s in the second term need not depend on each other; in other words

µ(O K ) = z | z 6 = 1 .

Exercise 52.2.3. Show that we always have that µ(O K ) comprises the roots to x n -1 for some integer n. (First, show it is a finite group under multiplication.)

We now quote, without proof, the so-called Dirichlet's unit theorem, which gives us a much more complete picture of what the units in O K are. Legend says that Dirichlet found the proof of this theorem during an Easter concert in the Sistine Chapel.

Theorem 52.2.4 (Dirichlet's unit theorem)

Let K be a number field with signature (r 1 , r 2 ) and set

s = r 1 + r 2 -1.

Then there exist units u 1 , . . . , u s such that every unit α ∈ O × K can be written uniquely in the form

α = ω • u n 1 1 . . . u ns s for ω ∈ µ(O K

) is a root of unity, and n 1 , . . . , n s ∈ Z.

More succinctly:

We have

O × K ∼ = Z r 1 +r 2 -1 × µ(O K ).

A choice of u 1 , . . . , u s is called a choice of fundamental units.

Here are some example applications.

Example 52.2.5 (Some unit groups) (a) Let K = Q(i) with signature (0, 1). Then we obtain s = 0, so Dirichlet's Unit theorem says that there are no units other than the roots of unity. Thus

O × K = {±1, ±i} where K = Q(i).

This is not surprising, since a

+ bi ∈ Z[i] is a unit if and only if a 2 + b 2 = 1. (b) Let K = Q( √), which has signature (2, 0). Then s = 1, so we expect exactly one fundamental unit. A fundamental unit is 2 + √ 3 (or 2 -√ 3, its inverse) with norm 1, and so we find

O × K = ±(2 + √ 3) n | n ∈ Z . (c) Let K = Q( 3 √

2) with signature (1, 1). Then s = 1, so we expect exactly one fundamental unit. The choice 1

+ 3 √ 2 + 3 √ 4. So O × K = ± 1 + 3 √ 2 + 3 √ 4 n | n ∈ Z .

I haven't actually shown you that these are fundamental units, and indeed computing fundamental units is in general hard.

§52.3 Finding fundamental units

Here is a table with some fundamental units.

d Unit d = 2 1 + √ 2 d = 3 2 + √ 3 d = 5 1 2 (1 + √ 5) d = 6 5 + 2 √ 6 d = 7 8 + 3 √ 7 d = 10 3 + √ 10 d = 11 10 + 3 √ 11

In general, determining fundamental units is computationally hard. However, once I tell you what the fundamental unit is, it's not too bad (at least in the case s = 1) to verify it. For example, suppose we want to show that 10 + 3

√ 11 is a fundamental unit of K = Q( √ 11), which has ring of integers Z[ √ 11].

If not, then for some n > 1, we would have to have

10 + 3 √ 11 = ± x + y √ 11 n .

For this to happen, at the very least we would need |y| < 3. We would also have x 2 -11y 2 = ±1. So one can just verify (using y = 1, 2) that this fails. The point is that: Since (10, 3) is the smallest (in the sense of |y|) integer solution to x 2 -11y 2 = ±1, it must be the fundamental unit. This holds more generally, although in the case that d ≡ 1 (mod 4) a modification must be made as x, y might be half-integers (like 

= Q( √ d) gives O K = Z[ 1 2 (1 + √ d)],

and so the fundamental unit may not actually be a solution. (For example, when d = 5, we get the solution ( 12 , 1 2 ).) Nonetheless, all integer solutions are eventually generated. To make this all concrete, here's a simple example.

Example 52.4.1 (x 2 -5y 2 = ±1)

Set K = Q( √ 5), so O K = Z[ 1 2 (1+ √ 5)]

. By Dirichlet's unit theorem, O × K is generated by a single element u. The choice

u = 1 2 + 1 2 √ 5

serves as a fundamental unit, as there are no smaller integer solutions to a 2 -5b 2 = ±4.

The first several powers of u are

n u n Norm -2 1 2 (3 - √ 5) 1 -1 1 2 (1 - √ 5) -1 0 1 1 1 1 2 (1 + √ 5) -1 2 1 2 (3 + √ 5) 1 3 2 + √ 5 -1 4 1 2 (7 + 3 √ 5) 1 5 1 2 (11 + 5 √ 5) -1 6 9 + 4 √ 5 1

One can see that the first integer solution is (2, 1), which gives -1. The first solution with +1 is (9, 4). Continuing the pattern, we find that every third power of u gives an integer solution (see also Problem 52B), with the odd ones giving a solution to

x 2 -5y 2 = -1 and the even ones a solution to x 2 -5y 2 = +1. All solutions are generated this way, up to ± signs (by considering ±u ±n ).

§52.5 A few harder problems to think about

Problem 52A (Fictitious account of the battle of Hastings). Determine the number of soldiers in the following battle:

The men of Harold stood well together, as their wont was, and formed thirteen squares, with a like number of men in every square thereof, and woe to the hardy Norman who ventured to enter their redoubts; for a single blow of Saxon war-hatched would break his lance and cut through his coat of mail . . . when Harold threw himself into the fray the Saxons were one might square of men, shouting the battle-cries, "Ut!", "Olicrosse!", "Godemite!" Problem 52B. Let d > 0 be a squarefree integer, and let u denote the fundamental unit of

Q( √ d). Show that either u ∈ Z[ √ d], or u n ∈ Z[ √ d] ⇐⇒ 3 | n.

Problem 52C. Show that there are no integer solutions to

x 2 -34y 2 = -1

despite the fact that -1 is a quadratic residue mod 34.

XIV Algebraic NT II: Galois and Ramification Theory

Part XIV: Contents 53 Things Galois . . 54.4 The Galois theory of finite fields . 

55

Ramification theory 55.1 Ramified / inert / split primes . 

56

The Frobenius element 56.1 Frobenius elements . 

57

Bonus: A Bit on Artin Reciprocity 57. 

. §53.1 Motivation

Prototypical example for this section:

Q( √ 2) and Q( 3 √ 2).

The key idea in Galois theory is that of embeddings, which give us another way to get at the idea of the "conjugate" we described earlier.

Let K be a number field. An embedding σ : K → C, is an injective field homomorphism: it needs to preserve addition and multiplication, and in particular it should fix 1.

Question 53.1.1. Show that in this context, σ(q) = q for any rational number q.

Example 53. 1.2 

(Examples of embeddings) (a) If K = Q(i), the two embeddings of K into C are z → z (the identity) and z → z (complex conjugation). (b) If K = Q( √ 2), the two embeddings of K into C are a + b √ 2 → a + b √ 2 (the identity) and a + b √ 2 → a -b √ 2 (conjugation). (c) If K = Q( 3 √ 2)

, there are three embeddings:

• The identity embedding, which sends 1 → 1 and 3 √ 2 → 3 √ 2.

• An embedding which sends 1 → 1 and 3 √ 2 → ω 3 √ 2, where ω is a cube root of unity. Note that this is enough to determine the rest of the embedding.

• An embedding which sends 1 → 1 and 3 √ 2 → ω 2 3 √ 2.

I want to make several observations about these embeddings, which will form the core ideas of Galois theory. Pay attention here! • First, you'll notice some duality between roots: in the first example, i gets sent to ±i, √ 2 gets sent to ± √ 2, and 3 √ 2 gets sent to the other roots of x 3 -2. This is no coincidence, and one can show this occurs in general. Specifically, suppose α has minimal polynomial

0 = c n α n + c n-1 α n-1 + • • • + c 1 α + c 0

where the c i are rational. Then applying any embedding σ to both sides gives

0 = σ(c n α n + c n-1 α n-1 + • • • + c 1 α + c 0 ) = σ(c n )σ(α) n + σ(c n-1 )σ(α) n-1 + • • • + σ(c 1 )σ(α) + σ(c 0 ) = c n σ(α) n + c n-1 σ(α) n-1 + • • • + c 1 σ(α) + c 0

where in the last step we have used the fact that c i ∈ Q, so they are fixed by σ. So, roots of minimal polynomials go to other roots of that polynomial.

• Next, I want to draw out a contrast between the second and third examples.

Specifically, in example (b) where we consider embeddings K = Q( √ 2) to C. The image of these embeddings lands entirely in K: that is, we could just as well have looked at K → K rather than looking at K → C. However, this is not true in (c): indeed Q( 3 √ 2) ⊂ R, but the non-identity embeddings have complex outputs!

The key difference is to again think about conjugates. Key observation:

The field K = Q( 3 √ 2)

is "deficient" because the minimal polynomial x 3 -2 has two other roots ω 3 √ 2 and ω 2 3 √ 2 not contained in K.

On the other hand K = Q( √ 2) is just fine because both roots of x 2 -2 are contained inside K. Finally, one can actually fix the deficiency in

K = Q( 3 √ 2) by completing it to a field Q( 3 √ 2, ω). Fields like Q(i) or Q( √ 2

) which are "self-contained" are called Galois extensions, as we'll explain shortly.

• Finally, you'll notice that in the examples above, the number of embeddings from K to C happens to be the degree of K. This is an important theorem, Theorem 53.3.1.

In this chapter we'll develop these ideas in full generality, for any field other than Q.

§53.2 Field extensions, algebraic extension, and splitting fields

Prototypical example for this section:

Q( 3 √ 2)/Q is an extension, C is

an algebraic extension of any number field.

First, we define a notion of one field sitting inside another, in order to generalize the notion of a number field. Definition 53.2.1. Let K and F be fields. If F ⊆ K, we write K/F and say K is a field extension of F .

Thus K is automatically an F -vector space (just like Q( √ 2) is automatically a Q-vector space). The degree is the dimension of this space, denoted [K : F ]. If [K : F ] is finite, we say K/F is a finite (field) extension.

That's really all. There's nothing tricky at all. 

Let F ⊆ K ⊆ L be fields with L/K, K/F finite. Then [L : K][K : F ] = [L : F ].

Proof. Basis bash: you can find a basis of L over K, and then expand that into a basis L over F . (Diligent readers can fill in details.)

Next, given a field (like Q( 3 √ 2)) we want something to embed it into (in our case C). So we just want a field that contains all the roots of all the polynomials. Let's agree that a field E is algebraically closed if every polynomial with coefficients in E is a product of linear polynomials in E, with the classic example is:

Example 53.2.4 (C) C is algebraically closed.

A major theorem is that any field F can be extended to an algebraically closed one F ; since all roots of polynomials in F [x] live in F , in particular so do all roots of polynomials in F [x]. Here is the result: Theorem 53.2.5 (Algebraic closures)

Any field F has algebraically closed field extensions. In fact, there is a unique such extension which is minimal by inclusion, called the algebraic closure and denoted F . (Here "minimal" means any other algebraically closed extension of F contains an isomorphic copy of F .) It has the property that every element of F is indeed the root of some polynomial with coefficients in F .

Example 53.2.6 (R = C = C ⊋ Q) C is the algebraic closure of R (and itself). But the algebraic closure Q of Q (i.e. the set of algebraic numbers) is a proper subfield of C (some complex numbers aren't the root of any rational-coefficient polynomial).

Usually we won't care much about what these extensions look like, and merely be satisfied they exist. Often we won't even use the algebraic closure, just any big enough field; for example, when working with a polynomial f with Q-coefficients, we simply consider roots of f as elements of C for convenience and concreteness, even though it may be less wasteful to use the smaller Q in place of C.

§53.3 Embeddings into algebraic closures for number fields

Now that I've defined all these ingredients, I can prove: Theorem 53.3.1 (The n embeddings of a number field)

Let K be a number field of degree n. Then there are exactly n field homomorphisms K → C, say σ 1 , . . . , σ n which fix Q.

Remark 53.3.2 -Note that a nontrivial homomorphism of fields is necessarily injective (the kernel is an ideal). This justifies the use of " →", and we call each σ i an embedding of K into C.

Proof. This is actually kind of fun! Recall that any irreducible polynomial over Q has distinct roots (Lemma 48. 1.2 ). We'll adjoin elements α 1 , α 2 , . . . , α m one at a time to Q, until we eventually get all of K, that is,

K = Q(α 1 , . . . , α n ).

Diagrammatically, this is It's common to see expressions like "let K be a number field of degree n, and σ 1 , . . . , σ n its n embeddings" without further explanation. The relation between these embeddings and the Galois conjugates is given as follows.

Q Q(α 1 ) Q(α 1 , α 2 ) . . . K C C C . . . C ⊃ ⊃ id ⊃ ⊃ τ 1 ⊃ ⊃ τ 2 ⊃ ⊃ τm=σ First,

Theorem 53.3.4 (Embeddings are evenly distributed over conjugates)

Let K be a number field of degree n with n embeddings σ 1 , . . . , σ n , and let α ∈ K have m Galois conjugates over Q.

Then σ j (α) is "evenly distributed" over each of these m conjugates: for any Galois conjugate β, exactly n m of the embeddings send α to β.

Proof. In the previous proof, adjoin α 1 = α first.

So, now we can define the trace and norm over Q in a nice way: given a number field K, we set

Tr K/Q (α) = n i=1 σ i (α) and N K/Q (α) = n i=1 σ i (α)

where σ i are the n embeddings of K into C.

§53.4 Everyone hates characteristic 2: separable vs irreducible

Prototypical example for this section: Q has characteristic zero, hence irreducible polynomials are separable. Now, we want a version of the above theorem for any field F . If you read the proof, you'll see that the only thing that ever uses anything about the field Q is Lemma 48. 1.2 , where we use the fact that

Irreducible polynomials over F have no double roots.

Let's call a polynomial with no double roots separable; thus we want irreducible polynomials to be separable. We did this for Q in the last chapter by taking derivatives. Should work for any field, right?

Nope. Suppose we took the derivative of some polynomial like 2x 3 + 24x + 9, namely 6x 2 + 24. In C it's obvious that the derivative of a nonconstant polynomial f ′ isn't zero. But suppose we considered the above as a polynomial in F 3 , i.e. modulo 3. Then the derivative is zero. Oh, no!

We have to impose a condition that prevents something like this from happening. We can get this to work for any field extension in which separability is not an issue.

Definition 53.4.6. A separable extension K/F is one where for each α ∈ K, the minimal polynomial of α over F is separable (for example, if F has characteristic zero).

A field F is perfect if any finite field extension K/F is separable.

In fact, as we see in the next chapter:

Theorem 53.4.7 (Finite fields are perfect)

Suppose F is a field with finitely many elements. Then it is perfect.

Thus, we will almost never have to worry about separability since every field we see in the Napkin is either finite or characteristic 0. So the inclusion of the word "separable" is mostly a formality.

Proceeding onwards, we obtain Theorem 53.4.8 (The n embeddings of any separable extension)

Let K/F be a separable extension of degree n and let F be an algebraic closure of F . Then there are exactly n field homomorphisms K → F , say σ 1 , . . . , σ n , which fix F .

In any case, this lets us define the trace for any separable normal extension.

Definition 53.4.9. Let K/F be a separable extension of degree n, and let σ 1 , . . . , σ n be the n embeddings into an algebraic closure of F . Then we define

Tr K/F (α) = n i=1 σ i (α) and N K/F (α) = n i=1 σ i (α).

When F = Q and the algebraic closure is C, this coincides with our earlier definition! §53.

5 Automorphism groups and Galois extensions

Prototypical example for this section:

Q( √ 2) is Galois but Q( 3 √ 2) is not.

We now want to get back at the idea we stated at the beginning of this section that

Q( 3 √ 2) is deficient in a way that Q( √ 2

) is not. First, we define the "internal" automorphisms.

Definition 53.5.1. Suppose K/F is a finite extension. Then Aut(K/F ) is the set of field isomorphisms σ : K → K which fix F . In symbols

Aut(K/F ) = {σ : K → K | σ is identity on F } .

This is a group under function composition!

Note that this time, we have a condition that F is fixed by σ. (This was not there before when we considered F = Q, because we got it for free.)

Example 53.5.2 (Old examples of automorphism groups)

Reprising the example at the beginning of the chapter in the new notation, we have: Here's a new example: let

(a) Aut(Q(i)/Q) ∼ = Z/2Z, with elements z → z and z → z. (b) Aut(Q( √ 2)/Q) ∼ = Z/2Z in the same way. (c) Aut(Q( 3 √ 2)/Q) is

K = Q( √ 2, √ 3). It turns out that Aut(K/Q) = {1, σ, τ, στ }, where σ : √ 2 → - √ 2 √ 3 → √ 3 and τ : √ 2 → √ 2 √ 3 → - √ 3.

In other words, Aut(K/Q) is the Klein Four Group.

First, let's repeat the proof of the observation that these embeddings shuffle around roots (akin to the first observation in the introduction):

Lemma 53.5.4 (Root shuffling in Aut(K/F )) Let f ∈ F [x],

suppose K/F is a finite extension, and assume α ∈ K is a root of f . Then for any σ ∈ Aut(K/F ), σ(α) is also a root of f .

Proof. Let f (x) = c n x n + c n-1 x n-1 + • • • + c 0 , where c i ∈ F . Thus, 0 = σ(f (α)) = σ (c n α n + • • • + c 0 ) = c n σ(α) n + • • • + c 0 = f (σ(α)).

In particular, taking f to be the minimal polynomial of α we deduce An embedding σ ∈ Aut(K/F ) sends an α ∈ K to one of its various Galois conjugates (over F ).

Next, let's look again at the "deficiency" of certain fields. Look at K = Q( 3 √ 2). So, again K/Q is deficient for two reasons. First, while there are three maps Q( 3 √ 2) → C, only one of them lives in Aut(K/Q), namely the identity. In other words, |Aut(K/Q)| is too small. Secondly, K is missing some Galois conjugates (ω 3 √ 2 and ω 2 3 √ 2). The way to capture the fact that there are missing Galois conjugates is the notion of a splitting field. In other words, the splitting field is the smallest field in which p(x) splits. 

2 -5 over Q is Q( √ 5)

. This is a degree 2 extension.

(b) The splitting field of

x 2 + x + 1 over Q is Q(ω),

where ω is a cube root of unity. This is a degree 2 extension.

(c) The splitting field of x 2 + 3x + 2 = (x + 1)(x + 2) is just Q! There's nothing to do.

Example 53.5.7 (Splitting fields: a cautionary tale)

The splitting field of x 3 -2 over Q is in fact

Q( 3 √ 2, ω)

and not just Q( 3 √ 2)! One must really adjoin all the roots, and it's not necessarily the case that these roots will generate each other.

To be clear:

• For x 2 -5, we adjoin √ 5 and this will automatically include -√ 5.

• For x 2 + x + 1, we adjoin ω and get the other root ω 2 for free.

• But for x 3 -2, if we adjoin 3 √ 2, we do NOT get ω 3 √ 2 and ω 2 3 √ 2 for free. Indeed,

Q( 3 √ 2) ⊂ R!

Note that in particular, the splitting field of x 3 -2 over Q is degree six, not just degree three.

In general, the splitting field of a polynomial can be an extension of degree up to n!. The reason is that if p(x) has n roots and no "coincidental" relations between them then any permutation of the roots will work. Now, we obtain:

Theorem 53.5.8 (Galois extensions are splitting)

For finite extensions K/F , |Aut(K/F )| divides [K : F ],

with equality if and only if K is the splitting field of some separable polynomial with coefficients in F .

The proof of this is deferred to an optional section at the end of the chapter. If K/F is a finite extension and |Aut(K/F )| = [K : F ], we say the extension K/F is Galois. In that case, we denote Aut(K/F ) by Gal(K/F ) instead and call this the Galois group of K/F . (b) The extension Q( √ 2, √ 3)/Q is Galois, since it's the splitting field of (x 2 -5) 2 -6 over Q. As discussed before, the Galois group is Z/2Z × Z/2Z.

(c) The extension Q( 3 √ 2)/Q is not Galois.

To explore Q( 3 √ 2) one last time:

Example 53.5.10 (Galois closures, and the automorphism group of

Q( 3 √ 2, ω))

Let's return to the field

K = Q( 3 √ 2, ω)

, which is a field with [K : Q] = 6. Consider the two automorphisms:

σ : 3 √ 2 → ω 3 √ 2 ω

→ ω and τ :

3 √ 2 → 3 √ 2 ω → ω 2 .

Notice that σ 3 = τ 2 = id. From this one can see that the automorphism group of K must have order 6 (it certainly has order ≤ 6; now use Lagrange's theorem). So,

K/Q is Galois! Actually one can check explicitly that Gal(K/Q) ∼ = S 3

is the symmetric group on 3 elements, with order 3! = 6.

This example illustrates the fact that given a non-Galois field extension, one can "add in" missing conjugates to make it Galois. This is called taking a Galois closure.

§53.6 Fundamental theorem of Galois theory

After all this stuff about Galois Theory, I might as well tell you the fundamental theorem, though I won't prove it. Basically, it says that if K/F is Galois with Galois group G, then:

Subgroups of G correspond exactly to fields E with F ⊆ E ⊆ K.

To tell you how the bijection goes, I have to define a fixed field.

Definition 53.6.1. Let K be a field and H a subgroup of Aut(K/F ). We define the fixed field of H, denoted K H , as In that case, the diagram of fields between Q and K matches exactly with the subgroups of G, as follows:

K H := {x ∈ K : σ(x) = x ∀σ ∈ H} .

Q( √ 2, √ 3) Q( √ 2) Q( √ 6) Q( √ 3) Q {id} {id, τ } {id, στ } {id, σ} G

We see that subgroups correspond to fixed fields. That, and much more, holds in general.

Theorem 53.6.4 (Fundamental theorem of Galois theory)

Let K/F be a Galois extension with Galois group G = Gal(K/F ).

(a) There is a bijection between field towers F ⊆ E ⊆ K and subgroups H ⊆ G:

             K | E | F              ⇐⇒              1 | H | G             

The bijection sends H to its fixed field K H , and hence is inclusion reversing. (c) K/E is always Galois, and its Galois group is Gal(K/E) = H.

(d) E/F is Galois if and only if H is normal in G. If so, Gal(E/F ) = G/H.

Exercise 53.6.5. Suppose we apply this theorem for

K = Q( 3 √ 2, ω).

Verify that the fact

E = Q( 3 √ 2)

is not Galois corresponds to the fact that S 3 does not have normal subgroups of order 2. Problem 53C (China Hong Kong Math Olympiad). Prove that there are no rational numbers p, q, r satisfying

cos 2π 7 = p + √ q + 3 √ r.

Problem 53D. Show that the only automorphism of R is the identity. Hence Aut(R) is the trivial group.

Problem 53E (Artin's primitive element theorem). Let K be a number field. Show that K ∼ = Q(γ) for some γ.

§53.8 (Optional) Proof that Galois extensions are splitting

We prove Theorem 53.5.8. First, we extract a useful fragment from the fundamental theorem.

Theorem 53.8.1 (Fixed field theorem)

Let K be a field and G a subgroup of Aut(K). Then [K :

K G ] = |G|.

The inequality itself is not difficult: The equality case is trickier.

The easier direction is when K is a splitting field. Assume K = F (α 1 , . . . , α n ) is the splitting field of some separable polynomial p ∈ F [x] with n distinct roots α 1 , . . . , α n . Adjoin them one by one: Let K/F be Galois, and p ∈ F [x] irreducible. If any root of p (in F ) lies in K, then all of them do, and in fact p is separable.

F F (α 1 ) F (α 1 , α 2 ) . . . K F F (α 1 ) F (α 1 , α 2 ) . . . K ⊃ id ⊃ τ 1 ⊃ τ 2 ⊃ τn=σ ⊃ ⊃ ⊃ ⊃Proof. Let α ∈ K be the prescribed root. Consider the set

S = {σ(α) | σ ∈ Gal(K/F )} . (Note that α ∈ S since Gal(K/F ) ∋ id.) By construction, any τ ∈ Gal(K/F ) fixes S. So if we construct p(x) = β∈S (x -β),

then by Vieta's Formulas, we find that all the coefficients of p are fixed by elements of σ.

By the equality case we specified in the exercise, it follows that p has coefficients in F ! (This is where we use the condition.) Also, by Lemma 53.5.4, p divides p.

Yet p was irreducible, so it is the minimal polynomial of α in F [x], and therefore we must have that p divides p. Hence p = p. Since p was built to be separable, so is p. Now we're basically done -pick a basis ω 1 , . . . , ω n of K/F , and let p i be their minimal polynomials; by the above, we don't get any roots outside K. Consider P = p 1 . . . p n , removing any repeated factors. The roots of P are ω 1 , . . . , ω n and some other guys in K. So K is the splitting field of P .

In this short chapter, we classify all fields with finitely many elements and compute the Galois groups. Nothing in here is very hard, and so most of the proofs are just sketches; if you like, you should check the details yourself.

The whole point of this chapter is to prove:

• A finite field F must have order p n , with p prime and n an integer.

• In this case, F has characteristic p.

• All such fields are isomorphic, so it's customary to use the notation F p n for "the" finite field of order p n if we only care up to isomorphism.

• The extension F/F p is Galois, and Gal(F/F p ) is a cyclic group of order n. The generator is the automorphism

σ : F → F by x → x p .

If you're in a hurry you can just remember these results and skip to the next chapter. §54. 

Let F ∼ = F 3 [X]/(X 2 + 1) ∼ = Z[i]/(3).

We can think of its elements as

{a + bi | 0 ≤ a, b ≤ 2} . Since (3) is prime in Z[i],

the ring of integers of Q(i), we see F is a field with 3 2 = 9 elements inside it. Note that, although this field has 9 elements, every element x has the property that 3x

= x + • • • + x 3 times = 0.

In particular, F has characteristic 3. §54.2 Finite fields have prime power order Lemma 54.2.1

If the characteristic of a field F isn't zero, it must be a prime number.

Proof.

Assume not, so n = ab for a, b < n. Then let

A = 1 F + • • • + 1 F a times ̸ = 0 and B = 1 F + • • • + 1 F b times ̸ = 0.

Then AB = 0, contradicting the fact that F is a field.

We like fields of characteristic zero, but unfortunately for finite fields we are doomed to have nonzero characteristic.

Lemma 54.2.2 (Finite fields have prime power orders)

Let F be a finite field. Then (a) Its characteristic is nonzero, and hence some prime p.

(b) The field F is a finite extension of F p , and in particular it is an F p -vector space.

(c) We have |F | = p n for some prime p, integer n.

Proof. Very briefly, since this is easy:

(a) Apply Lagrange's theorem (or pigeonhole principle!) to (F, +) to get the characteristic isn't zero.

(b) The additive subgroup of (F, +) generated by 1 F is an isomorphic copy of F p .

(c) Since it's a field extension, F is a finite-dimensional vector space over F p , with some basis e 1 , . . . , e n . It follows that there are p n elements of F .

Remark 54.2.3 -An amusing alternate proof of (c) by contradiction: if a prime q ̸ = p divides |F |, then by Cauchy's theorem (Problem 17A ⋆ ) on (F, +) there's a (nonzero) element x of order q. Evidently

x • (1 F + • • • + 1 F q times ) = 0

then, but x ̸ = 0, and hence the characteristic of F also divides q, which is impossible.

An important point in the above proof is that Lemma 54.2.4 (Finite fields are field extensions of F p )

If |F | = p n is a finite field, then there is an isomorphic copy of F p sitting inside F . Thus F is a field extension of F p .

We want to refer a lot to this copy of F p , so in what follows:

Abuse of Notation 54.2.5. Every integer n can be identified as an element of F , namely

n := 1 F + • • • + 1 F n times .

Note that (as expected) this depends only on n (mod p).

This notation makes it easier to think about statements like the following.

Theorem 54.2.6 (Freshman's dream)

For any a, b ∈ F we have (a + b) p = a p + b p .

Proof. Use the Binomial theorem, and the fact that p i is divisible by p for 0 < i < p.

Exercise 54.2.7. Convince yourself that this proof works.

§54.3 All finite fields are isomorphic

We next proceed to prove "Fermat's little theorem":

Theorem 54.3.1 (Fermat's little theorem in finite fields)

Let F be a finite field of order p n . Then every element x ∈ F satisfies

x p n -x = 0.

Proof. If x = 0 it's true; otherwise, use Lagrange's theorem on the abelian group (F, ×) to get x p n -1 = 1 F .

We can now prove the following result, which is the "main surprise" about finite fields: that there is a unique one up to isomorphism for each size.

Theorem 54.3.2 (Complete classification of finite fields)

A field F is a finite field with p n elements if and only if it is a splitting field of x p n -x over F p .

Proof.

By "Fermat's little theorem", all the elements of F satisfy this polynomial. So we just have to show that the roots of this polynomial are distinct (i.e. that it is separable).

To do this, we use the derivative trick again: the derivative of this polynomial is

p n • x p n -1 -1 = -1

which has no roots at all, so the polynomial cannot have any double roots.

Definition 54.3.3. For this reason, it's customary to denote the field with p n elements by F p n .

Note that the polynomial x p n -x (mod p) is far from irreducible, but the computation above shows that it's separable.

Example 54.3.4 (The finite field of order nine again)

The polynomial x 9 -x is separable modulo 3 and has factorization

x(x + 1)(x + 2)(x 2 + 1)(x 2 + x + 2)(x 2 + 2x + 2) (mod 3).

So if F has order 9, then we intuitively expect it to be the field generated by adjoining all the roots: 0, 1, 2, as well as ±i, 1 ± i, 2 ± i. Indeed, that's the example we had at the beginning of this chapter.

(Here i denotes an element of F 9 satisfying i 2 = -1. The notation is deliberately similar to the usual imaginary unit.) §54.

4 The Galois theory of finite fields

Retain the notation F p n now (instead of F like before). By the above theorem, it's the splitting field of a separable polynomial, hence we know that F p n /F p is a Galois extension. We would like to find the Galois group.

In fact, we are very lucky: it is cyclic. First, we exhibit one such element σ p ∈ Gal(F p n /F p ): Theorem 54.4.1 (The pth power automorphism)

The map σ p : F p n → F p n defined by σ p (x) = x p is an automorphism, and moreover fixes F p . This is called the Frobenius automorphism, and will re-appear later on in Chapter 56.

Proof. It's a homomorphism since it fixes 1, respects multiplication, and respects addition.

Question 54.4.2. Why does it respect addition? Next, we claim that it is injective. To see this, note that

x p = y p ⇐⇒ x p -y p = 0 ⇐⇒ (x -y) p = 0 ⇐⇒ x = y.

Here we have again used the Freshman's Dream. Since F p n is finite, this injective map is automatically bijective. The fact that it fixes F p is Fermat's little theorem.

We're very interested in how rational primes p factor in a bigger number field K. Some examples of this behavior: in Z[i] (which is a UFD!), we have factorizations

(2) = (1 + i) 2 (3) = (3) (5) = (2 + i)(2 -i).

In this chapter we'll learn more about how primes break down when they're thrown into bigger number fields. Using weapons from Galois Theory, this will culminate in a proof of Quadratic Reciprocity. §55.1 Ramified / inert / split primes Prototypical example for this section: In Z[i], 2 is ramified, 3 is inert, and 5 splits.

Let p be a rational prime, and toss it into O K . Thus we get a factorization into prime ideals

p • O K = p e 1 1 . . . p eg g .

We say that each p i is above (p). 1 Pictorially, you might draw this as follows:

K ⊃ O K p i Q ⊃ Z (p)

Some names for various behavior that can happen:

• We say p is ramified if e i > 1 for some i. For example 2 is ramified in Z[i].

• We say p is inert if g = 1 and e 1 = 1; i.e. (p) remains prime. For example 3 is inert in

Z[i].

• We say p is split if g > 1. For example 5 is split in Z[i].

Question 55.1.1. More generally, for a prime p in Z[i]:

• p is ramified exactly when p = 2.

• p is inert exactly when p ≡ 3 (mod 4).

• p is split exactly when p ≡ 1 (mod 4).

Prove this. §55.2 Primes ramify if and only if they divide ∆ K

The most unusual case is ramification: Just like we don't expect a randomly selected polynomial to have a double root, we don't expect a randomly selected prime to be ramified. In fact, the key to understanding ramification is the discriminant.

For the sake of discussion, let's suppose that K is monogenic,

O K = Z[θ]

, where θ has minimal polynomial f . Let p be a rational prime we'd like to factor. If f factors as f e 1 1 . . . f eg g , then we know that the prime factorization of (p) is given by

p • O K = i (p, f i (θ)) e i .

In particular, p ramifies exactly when f has a double root mod p! To detect whether this happens, we look at the polynomial discriminant of f , namely

∆(f ) = i<j (z i -z j ) 2

and see whether it is zero mod p -thus p ramifies if and only if this is true.

It turns out that the naïve generalization to any number field works if we replace ∆(f ) by just the discriminant ∆ K of K; (these are the same for monogenic O K by Problem 51C ⋆ ). That is, Theorem 55. 2 

.1 (Discriminant detects ramification)

Let p be a rational prime and K a number field. Then p is ramified if and only if p divides ∆ K .

Example 55.2.2 (Ramification in the Gaussian integers)

Let K = Q(i) so O K = Z[i] and ∆ K = -4.

As predicted, the only prime ramifying in Z[i] is 2, the only prime factor of ∆ K .

In particular, only finitely many primes ramify.

§55.3 Inertial degrees

Prototypical example for this section:

(7) has inertial degree 2 in Z[i] and (2 + i) has inertial degree 1 in Z[i].

Recall that we were able to define an ideal norm N(a) = |O K /a| measuring how "roomy" the ideal a is. For example, (5) has ideal norm 5 2 = 25 in Z[i], since 

Z[i]/(5) ∼ = {a + bi | a, b ∈ Z/5Z}

(b) Let (5) = (2 + i)(2 -i).

The inertial degrees of (2 + i) and (2 -i) are both 1. Indeed, Z[i]/(2 + i) only gives "one degree" of space, since each of its elements can be viewed as integers modulo 5, and there are only 5 1 = 5 elements.

If you understand this, it should be intuitively clear why the sum of e i f i should equal n. §55.4 The magic of Galois extensions OK, that's all fine and well. But something really magical happens when we add the additional hypothesis that K/Q is Galois: all the inertial degrees and ramification indices are equal. We set about proving this.

Let K/Q be Galois with G = Gal(K/Q). Note that if p ⊆ O K is a prime above p, then the image σ img (p) is also prime for any σ ∈ G (since σ is an automorphism!). Moreover, since p ∈ p and σ fixes Q, we know that p ∈ σ img (p) as well.

Thus, by the pointwise mapping, the Galois group acts on the prime ideals above a rational prime p. Picture:

p p 1 p 2 p 3 p 4 p 5 p 6 σ

The notation σ img (p) is hideous in this context, since we're really thinking of σ as just doing a group action, and so we give the shorthand: Abuse of Notation 55.4.1. Let σp be shorthand for σ img (p).

Since the σ's are all bijections (they are automorphisms!), it should come as no surprise that the prime ideals which are in the same orbit are closely related. But miraculously, it turns out there is only one orbit! Theorem 55. 4 .2 (Galois group acts transitively)

Let K/Q be Galois with G = Gal(K/Q). Let {p i } be the set of distinct prime ideals in the factorization of p • O K (in O K ).

Then G acts transitively on the p i : for every i and j, we can find σ such that σp i = p j .

In other words, All of the {p i } are Galois conjugates of each other.

Before proving this, let us consider the easier problem of factorization into elements.

Suppose O K is an UFD, and p factors as up 1 p 2 • • • p n in O K , where p i are irreducibles and u is an unit. Show that the p i are all conjugates of each other, up to multiplication by an unit. Proof. Let q = N K/Q (p 1 ) be the product of all conjugates of p 1 , then q ∈ Q. Thus p | q, so each p i is a factor of q, and we're done by unique factorization.

Unfortunately, the product of all conjugates of an ideal p 1 is not necessarily of the form p • O K (for example, K = Q[i] and (1 + i) has no other conjugates). So in the proof, we pick x which is an "representative" of p 1 .

Proof of Theorem 55. 4.2 . Because p i are distinct primes, by the Chinese remainder theorem, we can find an x ∈ O K such that x ≡ 0 (mod p 1 )

x ≡ 1 (mod p i ) for i ≥ 2 Then, compute the norm N K/Q (x) = σ∈Gal(K/Q) σ(x). Each σ(x) is in K because K/Q is Galois! Since N K/Q (x)

is an integer and divisible by p 1 , we should have that N K/Q (x) is divisible by p. Thus it should be divisible by p 2 as well. Thus, for some σ ∈ Gal(K/Q), σ(x) is divisible by p 2 , equivalently, x is divisible by σ -1 p 2 . But by the way we selected x, we have within the factors of p, x is divisible by only p 1 ! So σ -1 p 2 = p 1 , and we're done.

Theorem 55. 4 

.4 (Inertial degree and ramification indices are all equal)

Assume K/Q is Galois. Then for any rational prime p we have

p • O K = (p 1 p 2 . . . p g ) e

for some e, where the p i are distinct prime ideals with the same inertial degree f . Hence [K : Q] = ef g.

Proof.

To see that the inertial degrees are equal, note that each σ induces an isomorphism

O K /p ∼ = O K /σ(p).

Because the action is transitive, all f i are equal.

Exercise 55.4.5. Using the fact that σ ∈ Gal(K/Q), show that

σ img (p • O K ) = p • σ img (O K ) = p • O K .

So for every σ, we have that p

• O K = p e i i = (σp i ) e i .

Since the action is transitive, all e i are equal.

Let's see an illustration of this.

Example 55. 4 .6 (Factoring 5 in a Galois/non-Galois extension) Let p = 5 be a prime.

(a) Let E = Q( 3 √ 2). One can show that O E = Z[ 3 √ 2]

, so we use the Factoring Algorithm on the minimal polynomial x 3 -2. Since x 3 -2 ≡ (x -3)(x 2 + 3x + 9) (mod 5) is the irreducible factorization, we have that

(5) = (5, 3 √ 2 -3)(5, 3 √ 4 + 3 3 √ 2 + 9)

which have inertial degrees 1 and 2, respectively. The fact that this is not uniform reflects that E is not Galois.

(b) Now let K = Q( 3 √ 2, ω), which is the splitting field of x 3 -2 over Q; now K is Galois. It turns out that O K = Z[ε]

where ε is a root of t 6 + 3t 5 -5t 3 + 3t + 1.

(this takes a lot of work to obtain, so we won't do it here). Modulo 5 this has an irreducible factorization (x 2 + x + 2)(x 2 + 3x + 3)(x 2 + 4x + 1) (mod 5), so by the Factorization Algorithm, (5) = (5, ε 2 + ε + 2)(5, ε 2 + 3ε + 3)(5, ε 2 + 4ε + 1). This time all inertial degrees are 2, as the theorem predicts for K Galois.

§55.5 (Optional) Decomposition and inertia groups

Let p be a rational prime. Thus

p • O K = (p 1 . . . p g ) e

and all the p i have inertial degree f . Let p denote a choice of the p i .

We can look at both the fields O K /p and Z/p = F p . Naturally, since O K /p is a finite field we can view it as a field extension of F p . So we can get the diagram

K ⊃ O K p O K /p ∼ = F p f Q ⊃ Z (p) F p .

At the far right we have finite field extensions, which we know are really well behaved. So we ask:

How are Gal ((O K /p)/F p ) and Gal(K/Q) related?

First, every σ ∈ Gal(K/Q) induces an automorphism of O K , which induces a map

O K → O K /p by α → σ(α) (mod p).

For this to induce a map in Gal ((O K /p)/F p ), it's necessary that σ(p) ⊆ p. So, we consider the subset of automorphisms that fixes p: Definition 55.5.1. Let D p ⊆ Gal(K/Q) be the stabilizer of p, that is

D p := {σ ∈ Gal(K/Q) | σp = p} .

We say D p is the decomposition group of p.

Note that this definition is in fact equivalent to the set of σ such that σ(p) ⊆ p, because a field isomorphism fixes the ideal norm N(p). So there's a natural map

D p θ - → Gal ((O K /p)/F p )

by declaring θ(σ) to just be "σ (mod p)". The fact that σ ∈ D p (i.e. σ fixes p) ensures this map is well-defined. Surprisingly, every element of Gal ((O K /p)/F p ) arises this way from some field automorphism of K.

Theorem 55. 5 

.2 (Decomposition group and Galois group)

Define θ as above. Then

• θ is surjective, and • its kernel is a group of order e, the ramification index.

In particular, if p is unramified then

D p ∼ = Gal ((O K /p)/F p ).

(The proof is not hard, but a bit lengthy and in my opinion not very enlightening.)

If p is unramified, then taking modulo p gives

D p ∼ = Gal ((O K /p)/F p ).

But we know exactly what Gal ((O

K /p)/F p ) is! We already have O K /p ∼ = F p f , and the Galois group is Gal ((O K /p)/F p ) ∼ = Gal F p f /F p ∼ = ⟨x → x p ⟩ ∼ = Z/f Z. So D p ∼ = Z/f Z as well. Let's now go back to D p θ - → Gal ((O K /p) : F p ) .

The kernel of θ is called the inertia group and denoted I p ⊆ D p ; it has order e. This gives us a pretty cool sequence of subgroups {1} ⊆ I ⊆ D ⊆ G where G is the Galois group (I'm dropping the p-subscripts now). Let's look at the corresponding fixed fields via the Fundamental theorem of Galois theory. Picture:

p ⊂ O K ⊂ K {1} K I I K D D (p) ⊂ Z ⊂ Q G Ramify e Inert f

Split g TODO: the rest of this section is broken and commented out, needs fix (actually I'm not sure what the correct statement is, for that matter).

§55.6 Tangential remark: more general Galois extensions

All the discussion about Galois extensions carries over if we replace K/Q by some different Galois extension K/F . Instead of a rational prime p breaking down in O K , we would have a prime ideal p of F breaking down as 

p • O L = (P 1 . . . P g ) e in O L

(i) = i p = i p ≡ 1 (mod 4) -i p ≡ 3 (mod 4).

From this we see that Frob p is the identity when p ≡ 1 (mod 4) and Frob p is complex conjugation when p ≡ 3 (mod 4).

Note that we really only needed to compute Frob p on i. If this seems too good to be true, a philosophical reason is "freshman's dream" where (x + y) p ≡ x p + y p (mod p) (and hence mod p). So if σ satisfies the functional equation on generators, it satisfies the functional equation everywhere.

We also have an important lemma:

Lemma 56. This lemma allows us to tell the splitting behavior of p just by computing Frob p , which will later be seen in Lemma 56.4.1 and Section 56.6.iii.

Exercise 56. 1.4 . Prove this lemma as by using the fact that O K /p is the finite field of order f p , and the Frobenius element is just x → x p on this field.

Let us now prove the main theorem. This will only make sense in the context of decomposition groups, so readers which skipped that part should omit this proof.

Proof of existence of Frobenius element. The entire theorem is just a rephrasing of the fact that the map θ defined in the last section is an isomorphism when p is unramified. Picture:

G = Gal(K/Q) D p Frob p T T 2 T | T f = 1 θ(Frob p ) = T θ ∼ =

In here we can restrict our attention to D p since we need to have σ(α) ≡ 0 (mod p) when α ≡ 0 (mod p). Thus we have the isomorphism

D p θ - → Gal ((O K /p)/F p ) .

But we already know Gal ((O K /p)/F p ), according to the string of isomorphisms

Gal ((O K /p)/F p ) ∼ = Gal F p f /F p ∼ = ⟨T = x → x p ⟩ ∼ = Z/f Z.

So the unique such element is the pre-image of T under θ.

§56.2 Conjugacy classes

Now suppose p 1 and p 2 are two primes above an unramified rational prime p. Then we can define Frob p 1 and Frob p 2 . Since the Galois group acts transitively, we can select σ ∈ Gal(K/Q) be such that σ(p 1 ) = p 2 .

We claim that

Frob

p 2 = σ • Frob p 1 • σ -1 .

Note that this is an equation in G. Proof. We've used the fact that G = Gal(K/Q) is transitive to show that Frob p 1 and Frob p 2 are conjugate if they both lie above p; hence it's contained in some conjugacy class. So it remains to check that for any p, σ, we have σ • Frob p • σ -1 = Frob p ′ for some p ′ . For this, just take p ′ = σp. Hence the set is indeed a conjugacy class.

In summary,

Frob p is determined up to conjugation by the prime p from which p arises.

So even though the Gothic letters look scary, the content of Frob p really just comes from the more friendly-looking rational prime p.

Example 56.2.3 (Frobenius elements in

Q( 3 √ 2, ω))

With those remarks, here is a more involved example of a Frobenius map. Let

K = Q( 3 √ 2, ω) be the splitting field of t 3 -2 = (t - 3 √ 2)(t -ω 3 √ 2)(t -ω 2 3 √ 2).

Thus K/Q is Galois. We've seen in an earlier example that

O K = Z[ε]

where ε is a root of t 6 + 3t 5 -5t3 + 3t + 1.

Let's consider the prime 5 which factors (trust me here) as

(5) = (5,

ε 2 + ε + 2)(5, ε 2 + 3ε + 3)(5, ε 2 + 4ε + 1) = p 1 p 2 p 3 .

Note that all the prime ideals have inertial degree 2. Thus Frob p i will have order 2 for each i.

Note that

Gal(K/Q) = permutations of {

In this S 3 there are 3 elements of order two: fixing one root and swapping the other two. These correspond to each of Frob p 1 , Frobp 2 , Frob p 3 .

In conclusion, the conjugacy class {Frob p 1 , Frob p 2 , Frob p 3 } associated to (5) is the cycle type (•)(• •) in S 3 .

§56.3 Chebotarev density theorem

Natural question: can we represent every conjugacy class in this way? In other words, is every element of G equal to Frob p for some p?

Miraculously, not only is the answer "yes", but in fact it does so in the nicest way possible: the Frob p 's are "equally distributed" when we pick a random p.

Theorem 56.3.1 (Chebotarev density theorem over Q)

Let C be a conjugacy class of G = Gal(K/Q). The density of (unramified) primes p such that {Frob p | p above p} = C is exactly |C| / |G|. In particular, for any σ ∈ G there are infinitely many rational primes p with p above p so that Frob p = σ.

By density, I mean that the proportion of primes p ≤ x that work approaches |C| |G| as x → ∞. Note that I'm throwing out the primes that ramify in K. This is no issue, since the only primes that ramify are those dividing ∆ K , of which there are only finitely many.

In other words, if I pick a random prime p and look at the resulting conjugacy class, it's a lot like throwing a dart at G: the probability of hitting any conjugacy class depends just on the size of the class. G 37.5%

37.5%

C 1 C 2 C 3 18.75% C 4 6.25%

Remark 56.3.2 -Happily, this theorem (and preceding discussion) also works if we replace K/Q with any Galois extension K/F ; in that case we replace "p over p" with "P over p". In that case, we use N(p) ≤ x rather than p ≤ x as the way to define density.

§56.4 Example: Frobenius elements of cyclotomic fields

Let q be a prime, and consider L = Q(ζ q ), with ζ q a primitive qth root of unity. You should recall from various starred problems that

• ∆ L = ±q q-2 , • O L = Z[ζ q ], and • The map σ n : L → L by ζ q → ζ n q

is an automorphism of L whenever gcd(n, q) = 1, and depends only on n (mod q). In other words, the automorphisms of L/Q just shuffle around the qth roots of unity. In fact the Galois group consists exactly of the elements {σ n }, namely

Gal(L/Q) = {σ n | n ̸ ≡ 0 (mod q)}.

As a group,

Gal(L/Q) = (Z/qZ) × ∼ = Z/(q -1)Z.

This is surprisingly nice, because elements of Gal(L/Q) look a lot like Frobenius elements already. Specifically:

Lemma 56.4.1 (Cyclotomic Frobenius elements)

In the cyclotomic setting L = Q(ζ q ), let p be a rational unramified prime and p above it. Then Frob p = σ p .

Proof. Observe that σ p satisfies the functional equation (check on generators). Done by uniqueness.

Question 56.4.2. Conclude that a rational prime p splits completely in O L if and only if p ≡ 1 (mod q). §56.

5 Frobenius elements behave well with restriction

Let L/Q and K/Q be Galois extensions, and consider the setup

L ⊃ P Frob P ∈ Gal(L/Q) K ⊃ p Frob p ∈ Gal(K/Q) Q ⊃ (p)

Here p is above (p) and P is above p. We may define Frob p : K → K and Frob P : L → L and want to know how these are related.

Both maps Frob P and Frob p induce the power-of-p map in the corresponding quotient field, hence we would expect them to be naturally the same.

Theorem 56.5.1 (Restrictions of Frobenius elements)

Assume L/Q and K/Q are both Galois. Let P and p be unramified as above. Then Frob P ↾ K = Frob p , i.e. for every α ∈ K, Frob p (α) = Frob P (α).

Proof. First, K/Q is normal, so Frob P fixes the image of K, that is, Frob P ↾ K ∈ Gal(K/Q) is well-defined.

We have the natural map ϕ : O K → O L → O L /P, and the quotient map q : O K → O K /p. Since p ⊆ P ∩ O K ⊆ ker ϕ, it follows ϕ factors through q to give a natural field homomorphism

O K /p → O L /P.

Since a field homomorphism is injective, Frob P induces the power-of-p map on O L /P, and everything is commutative, the theorem follows.

In short, the point of this section is that

Frobenius elements upstairs restrict to Frobenius elements downstairs. §56.6 Application: Quadratic reciprocity

We now aim to prove: Theorem 56.6.1 (Quadratic reciprocity)

Let p and q be distinct odd primes. Then

p q q p = (-1) p-1 2 • q-1 2 .

(See, e.g. [Le] for an exposition on quadratic reciprocity, if you're not familiar with it.) §56.

6.i Step 1: Setup

For this proof, we first define

L = Q(ζ q )

where ζ q is a primitive qth root of unity. Then L/Q is Galois, with Galois group G.

Question 56.6.2. Show that G has a unique subgroup H of index two.

In fact, we can describe it exactly: viewing G ∼ = (Z/qZ) × , we have

H = {σ n | n quadratic residue mod q} .

By the fundamental theorem of Galois Theory, there ought to be a degree 2 extension of Q inside Q(ζ q ) (that is, a quadratic field). Call it Q( √ q * ), for q * squarefree:

L = Q(ζ q ) {1} K = Q( √ q * ) H Q G q-1 2 2

Exercise 56.6.3. Note that if a rational prime ℓ ramifies in K, then it ramifies in L. Use this to show that q * = ±q and q * ≡ 1 (mod 4).

Together these determine the value of q * .

(Actually, it is true in general ∆ K divides ∆ L in a tower L/K/Q.) §56.

6.ii Step 2: Reformulation

Now we are going to prove:

Theorem 56.6.4 (Quadratic reciprocity, equivalent formulation)

For distinct odd primes p, q we have p q = q * p .

Exercise 56.6.5. Using the fact that -1 p = (-1)

p- 12 , show that this is equivalent to quadratic reciprocity as we know it.

We look at the rational prime p in Z. Either it splits into two in K or is inert; either way let p be a prime factor in the resulting decomposition (so p is either p • O K in the inert case, or one of the primes in the split case). Then let P be above p. It could possibly also split in K: the picture looks like

O L = Z[ζ q ] ⊃ P Z[ζ q ]/P ∼ = F p f O K = Z[ 1+ √ q * 2 ] ⊃ p F p or F p 2 Z ⊃ (p) F p

Question 56.6.6. Why is p not ramified in either K or L? §56.

6.iii Step 3: Introducing the Frobenius

Now, we take the Frobenius

σ p = Frob P ∈ Gal(L/Q).

We claim that Frob P ∈ H ⇐⇒ p splits in K. In other words,

p q = 1 ⇐⇒ σ p ∈ H ⇐⇒ Frob P ∈ H ⇐⇒ Frob p = id K ⇐⇒ ord Frob p = 1 ⇐⇒ f p = 1 ⇐⇒ p splits in Z 1 2 (1 + q * ) ⇐⇒ q * p = 1.

This completes the proof.

§56.7 Frobenius elements control factorization

Prototypical example for this section: Frob p controlled the splitting of p in the proof of quadratic reciprocity; the same holds in general.

In the proof of quadratic reciprocity, we used the fact that Frobenius elements behaved well with restriction in order to relate the splitting of p with properties of Frob p .

In fact, there is a much stronger statement for any intermediate field Q ⊆ E ⊆ K which works even if E/Q is not Galois. It relies on the notion of a factorization pattern.

Here is how it goes.

Set n = [E : Q], and let p be a rational prime unramified in K. Then p can be broken in E as p • O E = p 1 p 2 . . . p g with inertial degrees f 1 , . . . , f g : (these inertial degrees might be different since E/Q isn't Galois). The numbers f 1 + • • • + f g = n form a partition of the number n. For example, in the quadratic reciprocity proof we had n = 2, with possible partitions 1 + 1 (if p split) and 2 (if p was inert). We call this the factorization pattern of p in E.

Next, we introduce a Frobenius Frob P above (p), all the way in K; this is an element of G = Gal(K/Q). Then let H be the group corresponding to the field E. Diagram:

K {1} Frob P E H p 1 . . . p g f 1 + • • • + f g = n Q G (p) n n

Then Frob P induces a permutation of the n left cosets gH by left multiplication (after all, Frob P is an element of G too!). Just as with any permutation, we may look at the resulting cycle decomposition, which has a natural "cycle structure": a partition of n.

g 1 H g 2 H g 3 H ×g ×g ×g 3 g = Frob P g 4 H g 5 H g 6 H g 7 H ×g ×g ×g ×g 4 n = 7 = 3 + 4

The theorem is that these coincide: Theorem 56.7.1 (Frobenius elements control decomposition)

Let Q ⊆ E ⊆ K an extension of number fields and assume K/Q is Galois (though E/Q need not be). Pick an unramified rational prime p; let G = Gal(K/Q) and H the corresponding intermediate subgroup. Finally, let P be a prime above p in K.

Then the factorization pattern of p in E is given by the cycle structure of Frob P acting on the left cosets of H.

Often, we take E = K, in which case this is just asserting that the decomposition of the prime p is controlled by a Frobenius element over it.

Sketch of Proof. Let α be an algebraic integer and f its minimal polynomial (of degree n).

Set E = Q(α) (which has degree n over Q). Suppose we're lucky enough that O E = Z[α], i.e. that E is monogenic. Then we know by the Factoring Algorithm, to factor any p in E, all we have to do is factor f modulo p, since if f = f e 1 1 . . . f eg g (mod p) then we have

(p) = i p i = i (f i (α), p) e i .

This gives us complete information about the ramification indices and inertial degrees; the e i are the ramification indices, and deg f i are the inertial degrees (since

O E /p i ∼ = F p [X]/(f i (X))).

In particular, if p is unramified then all the e i are equal to 1, and we get

n = deg f = deg f 1 + deg f 2 + • • • + deg f g .

Once again we have a partition of n; we call this the factorization pattern of f modulo p. So, to see the factorization pattern of an unramified p in O E , we just have to know the factorization pattern of f (mod p).

To prove our theorem, we will show that the factorization pattern of f (mod p) corresponds exactly to the cycle decomposition of the action of Frob P on the roots of f and that the roots of f correspond exactly to the cosets of H in G.

To do this, suppose S = {α 1 , α 2 , . . . , α n } are the roots of f (distinct roots since f is irreducible over Q). We let Frob P act on S. This splits S into orbits S 1 , S 2 , . . . , S k .

Construct polynomials f i with coefficients in E having roots exactly the elements of S i . This forms a factorization of f over E, say

f = f 1 f 2 . . . f k .

We claim that this in fact induces a factorization of f (mod p). To see this, consider the images of these polynomials f i under the quotient O K → O K /P, denote them by f i . Then since p is unramified, we know that the decomposition group D(P|p) is isomorphic to the Galois group G = Gal((O E /P)/(Z/pZ)). Thus Frob P corresponds to the generator σ of G. It is not hard to believe that the action of Frob P on the roots of f is the same as that of σ on the roots of f . Since the roots of f i form an orbit under the action of Frob P , we see that the roots of f i form an orbit under the action of σ and hence under the action of G. It is now a standard fact of Galois theory that f i is an irreducible polynomial over F p (since it is fixed by G), thus the claim is proved. Now we just need to observe that the roots of f correspond to the cosets of H, this will be established later.

We saw above that given the factorization pattern of f (mod p), we can determine the factorization pattern of an unramified prime p in O E .

Turning this on its head, if we want to know the factorization pattern of f (mod p), we just need to know how p decomposes. And it turns out these coincide even without the assumption that E is monogenic.

Theorem 56.7.2 (Frobenius controls polynomial factorization)

Let α be an algebraic integer with minimal polynomial f , and let E = Q(α). Then for any prime p unramified in the splitting field K of f , the following coincide:

(i) The factorization pattern of p in E.

(ii) The factorization pattern of f (mod p).

(iii) The cycle structure associated to the action of Frob P ∈ Gal(K/Q) on the roots of f , where P is above p in K.

Example 56.7.3 (Factoring x 3 -2 (mod 5))

Let α = 3 √ 2 and f = x 3 -2, so E = Q( 3 √ 2)

. Set p = 5 and finally, let K = Q( 3 √ 2, ω) be the splitting field. Setup:

K = Q( 3 √ 2, ω) P x 3 -2 = (x -3 √ 2)(x -3 √ 2ω)(x -3 √ 2ω 2 ) E = Q( 3 √ 2) p x 3 -2 = (x -3 √ 2)(x 2 + 3 √ 2x + 3 √ 4) Q (5) x 3 -2 irreducible over Q 2 3

The three claimed objects now all have shape 2 + 1:

(i) By the Factoring Algorithm, we have (5) = (5, 3 √ 2 -3)(5, 9 + 3 3 √ 2 + 3 √ 4).

(ii) We have x 3 -2 ≡ (x -3)(x 2 + 3x + 9) (mod 5).

(iii) We saw before that Frob P = (•)(• •).

Sketch of Proof.

Letting n = deg f . Let H be the subgroup of G = Gal(K/Q) corresponding to E, so |G/H| = n. Pictorially, we have

K {1} P E = Q(α) H p Q G (p)

We claim that (i), (ii), (iii) are all equivalent to (iv) The pattern of the action of Frob P on the G/H.

In other words we claim the cosets correspond to the n roots of f in K. Indeed H is just the set of τ ∈ G such that τ (α) = α, so there's a bijection between the roots and the cosets G/H by τ H → τ (α). Think of it this way: if G = S n , and H = {τ : τ (1) = 1}, then G/H has order n!/(n -1)! = n and corresponds to the elements {1, . . . , n}. So there is a natural bijection from (iii) to (iv).

The fact that (i) is in bijection to (iv) was the previous theorem, Theorem 56.7. 1 . The correspondence (i) ⇐⇒ (ii) is a fact of Galois theory, so we omit the proof here.

All this can be done in general with Q replaced by F ; for example, in [Le02] .

§56.8 Example application: IMO 2003 problem 6

As an example of the power we now have at our disposal, let's prove: Problem 6. Let p be a prime number. Prove that there exists a prime number q such that for every integer n, the number n p -p is not divisible by q.

We will show, much more strongly, that there exist infinitely many primes q such that X p -p is irreducible modulo q.

Solution. Okay! First, we draw the tower of fields Hence by Cauchy's theorem (Problem 17A ⋆ , which is a purely group-theoretic fact) we can find a σ ∈ G of order p. By Chebotarev, there exist infinitely many rational (unramified) primes q ̸ = p and primes Q ⊆ O K above q such that Frob Q = σ. (Yes, that's an uppercase Gothic Q. Sorry.) We claim that all these q work. By Theorem 56.7.2, the factorization of f (mod q) is controlled by the action of σ = Frob Q on the roots of f . But σ has prime order p in G! So all the lengths in the cycle structure have to divide p. Thus the possible factorization patterns of f are

Q ⊆ Q( p √ p) ⊆ K where K is the splitting field of f (x) = x p -p. Let E = Q( p √ p)

p = 1 + 1 + • • • + 1 p times or p = p.

So we just need to rule out the p = 1 + • • • + 1 case now: this only happens if f breaks into linear factors mod q. Intuitively this edge case seems highly unlikely (are we really so unlucky that f factors into linear factors when we want it to be irreducible?). And indeed this is easy to see: this means that σ fixes all of the roots of f in K, but that means σ fixes K altogether, and hence is the identity of G, contradiction. Problem 56D. Let n be an odd integer which is not a prime power. Show that the nth cyclotomic polynomial is not irreducible modulo any rational prime.

Problem 56E (Putnam 2012 B6). Let p be an odd prime such that p ≡ 2 (mod 3). Let π be a permutation of F p by π(x) = x 3 (mod p). Show that π is even if and only if p ≡ 3 (mod 4).

57 Bonus: A Bit on Artin Reciprocity

In this chapter, I'm going to state some big theorems of global class field theory and use them to deduce the Kronecker-Weber plus Hilbert class fields. No proofs, but hopefully still appreciable. For experts: this is global class field theory, without ideles.

Here's the executive summary: let K be a number field. Then all abelian extensions L/K can be understood using solely information intrinsic to K: namely, the ray class groups (generalizing ideal class groups).

§57.1 Overview

At the end of this section, for an Abelian field extension L/K, we will define the Artin symbol L/K p , which generalizes the Legendre symbol a p :

• Above the solidus, instead of an integer a, we have a field extension L/K.

• Below the solidus, instead of a rational prime p, we have a prime ideal p of K.

We require p to not ramify in the extension L/K for the symbol to be defined. And, at the end, we want to state the Artin reciprocity theorem, which looks something like the following:

For primes p, L/K p depends only on "p (mod f)".

Here, f is a "modulus", which only depends on the field extension L/K. In order to do that, we first need to define what it means for two ideals to be coprime modulo something. We will divide up the ideals of O K that is "coprime" to f into "residue classes modulo f" (we will call them "ray classes" from now on) in such a way that:

• It generalizes the class group -two ideals that belong to different ideal classes (i.e.

are nonisomorphic as O K -modules) belong to different ray classes.

• It respects the multiplicative structure -if p is in the same ray class as p ′ , and q is in the same ray class as q ′ , then pq is in the same ray class as p ′ q ′ .

Note that there is no analogue of element addition for the ideals (for instance, (1) = (-1) but ( 1) + (1) ̸ = (1) + (-1)), so this is the best we can hope for.

In other words, the ray classes will form an abelian group under multiplication, with the operation induced from ideal multiplication.

• For a fixed modulus f, there are only finitely many ray classes.

In the section above, you may think of a prime ideal p ∈ O K as an irreducible factor, such that all ideals can be written as products of. However, they can also naturally be used as a modulus:

A prime p gives a way to divide the elements of O K into residue classes that respects the addition and multiplication of elements.

This can further be generalized to divide up the ideals of O K into ray classes -unfortunately, using only the finite primes is insufficient to divide up the ideals the way we want, as later seen in Example 57.3.3. So, the infinite primes will be introduced in order to divide up the elements, as well as the ideals, into classes that satisfies the multiplicative structure.

§57.2 Infinite primes

Prototypical example for this section: Q( √ -5) has a complex infinite prime, Q( √ 5) has two real infinite ones.

Let K be a number field of degree n and signature (r, s). We know what a prime ideal of O K is, but we now allow for the so-called infinite primes, which I'll describe using the embeddings. 1 Recall there are n embeddings σ : K → C, which consist of • r real embeddings where im σ ⊆ R, and • s pairs of conjugate complex embeddings.

Hence r + 2s = n. The first class of embeddings form the real infinite primes, while the complex infinite primes are the second type. We say K is totally real (resp totally complex) if all its infinite primes are real (resp complex).

Example 57.2.1 (Examples of infinite primes)

• Q has a single real infinite prime. We often write it as ∞.

• Q( √ -5) has a single complex infinite prime, and no real infinite primes. Hence totally complex.

• Q( √ 5) has two real infinite primes, and no complex infinite primes. Hence totally real.

§57.3 Modular arithmetic with infinite primes

A modulus (or module) of K is a formal product m = p p ν(p)

where the product runs over all primes, finite and infinite. (Here ν(p) is a nonnegative integer, of which only finitely many are nonzero.) We also require that • ν(p) = 0 for any complex infinite prime p, and • ν(p) ≤ 1 for any real infinite prime p.

Obviously, every m can be written as m = m 0 m ∞ by separating the finite from the (real) infinite primes.

We • Let I K (m) denote the set of all fractional ideals of K which are relatively prime to m, which is an abelian group.

• Let P K (m) be the subgroup of I K (m) generated by

αO K | α ∈ K × and α ≡ 1 (mod m) .

This is sometimes called a "ray" of principal ideals.2 

Finally define the ray class group of m to be

C K (m) = I K (m)/P K (m).

This group is known to always be finite. Note the usual class group is C K (1).

One last definition that we'll use right after Artin reciprocity:

Definition 57.3.2. A congruence subgroup of m is a subgroup H with

P K (m) ⊆ H ⊆ I K (m).

Thus C K (m) is a group which contains a lattice of various quotients I K (m)/H, where H is a congruence subgroup.

This definition takes a while to get used to, so here are examples.

Example 57.3.3 (Ray class groups in Q, finite modulus)

Consider K = Q with infinite prime ∞. Then • If we take m = 1 then I Q (1

) is all fractional ideals, and P Q (1) is all principal fractional ideals. Their quotient is the usual class group of Q, which is trivial.

• Now take m = 8. Thus

I Q (8) ∼ = a b Z | a/b ≡ 1, 3, 5, 7 (mod 8) . Moreover P Q (8) ∼ = a b Z | a/b ≡ 1 (mod 8) .

You might at first glance think that the quotient is thus (Z/8Z) × . But the issue is that we are dealing with ideals: specifically, we have 7Z = -7Z ∈ P Q (8) because -7 ≡ 1 (mod 8). So actually, we get

C Q (8) ∼ = {1, 3, 5, 7 mod 8} / {1, 7 mod 8} ∼ = (Z/4Z) × . More generally, C Q (m) = (Z/mZ) × /{±1}.

Example 57.3.4 (Ray class groups in Q, infinite moduli)

Consider K = Q with infinite prime ∞ again. • Now take m = ∞. As before I Q (∞) = Q × . Now

, by definition we have

P Q (∞) = a b Z | a/b > 0 .

At first glance you might think this was Q >0 , but the same behavior with ideals shows in fact P Q (∞) = Q × . So in this case, P Q (∞) still has all principal fractional ideals. Therefore, C Q (∞) is still trivial.

• Finally, let m = 8∞. As before

I Q (8∞) ∼ = a b Z | a/b ≡ 1, 3

, 5, 7 (mod 8) . Now in this case:

P Q (8∞) ∼ = a b Z | a/b ≡ 1 (mod 8) and a/b > 0 .

This time, we really do have -7Z / ∈ P Q (8∞): we have 7 ̸ ≡ 1 (mod 8) and also -7 < 0. So neither of the generators of 7Z are in P Q (8∞). Thus we finally obtain C Q (8∞) ∼ = {1, 3, 5, 7 mod 8} / {1 mod 8} ∼ = (Z/8Z) × with the bijection C Q (8∞) → (Z/8Z) × given by aZ → |a| (mod 8).

More generally, C Q (m∞) = (Z/mZ) × . §57.4 Infinite primes in extensions I want to emphasize that everything above is intrinsic to a particular number field K.

After this point we are going to consider extensions L/K but it is important to keep in mind the distinction that the concept of modulus and ray class group are objects defined solely from K rather than the above L. Now take a Galois extension L/K of degree m. We already know prime ideals p of K break into a product of prime ideals P of L in a nice way, so we want to do the same thing with infinite primes. This is straightforward: each of the n infinite primes σ : K → C lifts to m infinite primes τ : L → C, by which I mean the diagram

L C K τ ⊃ σ commutes.

Hence like before, each infinite prime σ of K has m infinite primes τ of L which lie above it.

For a real prime σ of K, if any of the resulting τ above it are complex, we say that the prime σ ramifies in the extension L/K. Otherwise it is unramified in L/K. An infinite prime of K is always unramified in L/K. In this way, we can talk about an unramified Galois extension L/K: it is one where all primes (finite or infinite) are unramified.

Example 57.4.1 (Ramification of ∞)

Let ∞ be the real infinite prime of Q.

• ∞ is ramified in Q( √ -5)/Q. • ∞ is unramified in Q( √ 5)/Q.

Note also that if K is totally complex then any extension L/K is unramified.

§57.5 Frobenius element and Artin symbol

Recall the key result:

Theorem 57.5.1 (Frobenius element)

Let L/K be a Galois extension. If p is a prime unramified in L/K, and P a prime above it in L, then there is a unique element of Gal(L/K), denoted Frob P , obeying

Frob P (α) ≡ α N(p) (mod P) ∀α ∈ O L .

Recall some examples from Example 56. 1.2 

Let L = Q(i), K = Q. We have Gal(L/K) ∼ = Z/2Z.

If p is an odd prime with P above it, then Frob P is the unique element such that (a + bi) p ≡ Frob P (a + bi) (mod P)

in Z[i].

In particular,

Frob P (i) = i p = i p ≡ 1 (mod 4) -i p ≡ 3 (mod 4).

From this we see that Frob P is the identity when p ≡ 1 (mod 4) and Frob P is complex conjugation when p ≡ 3 (mod 4). where p > 2 is prime. Consider the element

Q( √ 2)/Q pZ ∈ Gal(Q( √ 2)/Q).

It is uniquely determined by where it sends √ 2. But in fact we have

Q( √ 2)/Q pZ √ 2 ≡ √ 2 p ≡ 2 p-1 2 • √ 2 ≡ 2 p √ 2 (mod P)

where 2 p is the usual Legendre symbol, and P is above p in Q( √ 2). Thus the Artin symbol generalizes the quadratic Legendre symbol.

Example 57.5.8 (Cubic Legendre symbol subsumed by Artin symbol) Similarly, it also generalizes the cubic Legendre symbol. To see this, assume θ is a primary prime in

K = Q( √ -3) = Q(ω) (thus O K = Z[ω]

is the Eisenstein integers). Then for example

K( 3 √ 2)/K θO K 3 √ 2 ≡ 3 √ 2 N(θ) ≡ 2 N(θ)-1 3 • 3 √ 2 ≡ 2 θ 3 3 √ 2. (mod P) where P is above (θ) in K( 3 √ 2).

§57.6 Artin reciprocity

Now, we further capitalize on the fact that Gal(L/K) is abelian. For brevity, in what follows let Ram(L/K) denote the primes of K (either finite or infinite) which ramify in L.

Definition 57.6.1. Let L/K be an abelian extension and let m be divisible by every prime in Ram(L/K). Then since L/K is abelian we can extend the Artin symbol multiplicatively to a map

L/K • : I K (m) ↠ Gal(L/K).

This is called the Artin map, and it is surjective (for example by Chebotarev Density).

Let H(L/K, m) ⊆ I K (m) denote the kernel of this map, so

Gal(L/K) ∼ = I K (m)/H(L/K, m).

We can now present the long-awaited Artin reciprocity theorem.

Theorem 57.6.2 (Artin reciprocity)

Let L/K be an abelian extension. Then there is a modulus f = f(L/K), divisible by exactly the primes of Ram(L/K), such that: for any modulus m divisible by all primes of Ram(L/K), we have

P K (m) ⊆ H(L/K, m) ⊆ I K (m) if and only if f | m.

We call f the conductor of L/K.

So the conductor f plays a similar role to the discriminant (divisible by exactly the primes which ramify), and when m is divisible by the conductor, H(L/K, m) is a congruence subgroup.

Here's the reason this is called a "reciprocity" theorem. The above theorem applies on m = f tells us P K (f) ⊆ H(L/K, f), so the Artin map factors through the quotient map

I K (f) ↠ I K (f)/P K (f). Recalling that C K (f) = I K (f)/P K (f)

, we get a sequence of maps

I K (f) C K (f) Gal(L/K) I K (f)/H(L/K, f) L/K • ∼ = Consequently:

For primes p ∈ I K (f), L/K p depends only on "p (mod f)".

Let's see how this result relates to quadratic reciprocity.

Example 57.6.3 (Artin reciprocity implies quadratic reciprocity)

The big miracle of quadratic reciprocity states that: for a fixed (squarefree) a, the Legendre symbol a p should only depend the residue of p modulo something. Let's see why Artin reciprocity tells us this a priori.

Let L = Q( √ a), K = Q. Then we've already seen that the Artin symbol

Q( √ a)/Q •

is the correct generalization of the Legendre symbol. Thus, Artin reciprocity tells us that there is a conductor

f = f(Q( √ a)/Q) such that Q( √ a)/Q p

depends only on the residue of p modulo f, which is what we wanted.

Here is an example along the same lines.

Example 57.6.4 (Cyclotomic field)

Let ζ be a primitive mth root of unity. For primes p, we know that Frob p ∈ Gal(Q(ζ)/Q) is "exactly" p (mod m). Let's translate this idea into the notation of Artin reciprocity.

We are going to prove

H(Q(ζ)/Q, m∞) = P Q (m∞) = a b Z | a/b ≡ 1 (mod m) .

This is the generic example of achieving the lower bound in Artin reciprocity. It also implies that

f(Q(ζ)/Q) | m∞.

It's well-known Q(ζ)/Q is unramified outside finite primes dividing m, so that the Artin symbol is defined on I K (m). Now the Artin map is given by

I Q (m) Gal(Q(ζ)/Q) (Z/mZ) × p (x → x p ) p (mod m). Q(ζ)/Q • ∼ =

So we see that the kernel of this map is trivial, i.e. it is given by the identity of the Galois group, corresponding to 1 (mod m). On the other hand, we've also computed P Q (m∞) already, so we have the desired equality.

In fact, we also have the following "existence theorem": every congruence subgroup appears uniquely once we fix m.

Theorem 57.6.5 (Takagi existence theorem) Fix K and let m be a modulus. Consider any congruence subgroup H, i.e.

P K (m) ⊆ H ⊆ I K (m).

Then H = H(L/K, m) for a unique abelian extension L/K.

Finally, such subgroups reverse inclusion in the best way possible: Lemma 57.6.6 (Inclusion-reversing congruence subgroups) Fix a modulus m. Let L/K and M/K be abelian extensions and suppose m is divisible by the conductors of L/K and M/K. Then

L ⊆ M if and only if H(M/K, m) ⊆ H(L/K, m).

Here by L ⊆ M we mean that L is isomorphic to some subfield of M .

Sketch of proof.

Let us first prove the equivalence with m fixed. In one direction, assume L ⊆ M ; one can check from the definitions that the diagram

I K (m) Gal(M/K) Gal(L/K) M/K • L/K •

commutes, because it suffices to verify this for prime powers, which is just saying that Frobenius elements behave well with respect to restriction. Then the inclusion of kernels follows directly. The reverse direction is essentially the Takagi existence theorem.

Note that we can always take m to be the product of conductors here.

To finish, here is a quote from Emil Artin on his reciprocity law:

I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define L-series for non-abelian extensions. But for them to agree with the L-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I called it the General Reciprocity Law and tried to prove it but couldn't, even after many tries. Then I showed it to the other number theorists, but they all laughed at it, and I remember Hasse in particular telling me it couldn't possibly be true.

Still, I kept at it, but nothing I tried worked. Not a week went by -for three years! -that I did not try to prove the Reciprocity Law. It was discouraging, and meanwhile I turned to other things. Then one afternoon I had nothing special to do, so I said, 'Well, I try to prove the Reciprocity Law again.' So I went out and sat down in the garden. You see, from the very beginning I had the idea to use the cyclotomic fields, but they never worked, and now I suddenly saw that all this time I had been using them in the wrong wayand in half an hour I had it. §57.7 A few harder problems to think about Problem 57A † (Kronecker-Weber theorem). Let L be an abelian extension of Q. Then L is contained in a cyclic extension Q(ζ) where ζ is an mth root of unity (for some m).

Problem 57B † (Hilbert class field). Let K be any number field. Then there exists a unique abelian extension E/K which is unramified at all primes (finite or infinite) and such that

• E/K is the maximal such extension by inclusion.

• Gal(E/K) is isomorphic to the class group of K.

• A prime p of K splits completely in E if and only if it is a principal ideal of O K .

We call E the Hilbert class field of K.

XV Algebraic Topology I: Homotopy

Part XV: Contents 58 Some topological constructions 58. 

59

Fundamental groups 59.1 Fusing paths together . 

60

Covering projections 60.1 Even coverings and covering projections . 

Example 58.2.2 (Interval modulo endpoints)

Suppose we take D 1 = [-1, 1] and quotient by the equivalence relation which identifies the endpoints -1 and 1. (Formally, x ∼ y ⇐⇒ (x = y) or {x, y} = {-1, 1}.) In that case, we simply recover S 1 :

-1 1 D 1 S 1 ≈ D 1 /∼ -1 ∼ 1

Observe that a small open neighborhood around -1 ∼ 1 in the quotient space corresponds to two half-intervals at -1 and 1 in the original space D 1 . This should convince you the definition we gave is the right one.

Example 58.2.3 (More quotient spaces)

Convince yourself that: • Generalizing the previous example, D n modulo its boundary S n-1 is S n .

• Given a square ABCD, suppose we identify segments AB and DC together.

Then we get a cylinder. (Think elementary school, when you would tape up pieces of paper together to get cylinders.) • In the previous example, if we also identify BC and DA together, then we get a torus. (Imagine taking our cylinder and putting the two circles at the end together.)

• Let X = R, and let x ∼ y if y -x ∈ Z. Then X/∼ is S 1 as well.

One special case that we did above: Definition 58.2.4. Let A ⊆ X. Consider the equivalence relation which identifies all the points of A with each other while leaving all remaining points inequivalent. (In other words, x ∼ y if x = y or x, y ∈ A.) Then the resulting quotient space is denoted X/A.

So in this notation,

D n /S n-1 = S n .

Abuse of Notation 58.2.5. Note that I'm deliberately being sloppy, and saying "D n /S n-1 = S n " or "D n /S n-1 is S n ", when I really ought to say "D n /S n-1 is homeomorphic to S n ". This is a general theme in mathematics: objects which are homoeomorphic/isomorphic/etc. are generally not carefully distinguished from each other.

§58.3 Product topology

Prototypical example for this section: R × R is R 2 , S 1 × S 1 is the torus. More interesting is the wedge sum, where two topological spaces X and Y are fused together only at a single base point.

Definition 58.4.3. Let X and Y be topological spaces, and x 0 ∈ X and y 0 ∈ Y be points. We define the equivalence relation ∼ by declaring x 0 ∼ y 0 only. Then the wedge sum of two spaces is defined as

X ∨ Y = (X ⨿ Y )/∼. Example 58.4.4 (S 1 ∨ S 1 is a figure eight)

Let X = S 1 and Y = S 1 , and let x 0 ∈ X and y 0 ∈ Y be any points. Then X ∨ Y is a "figure eight": it is two circles fused together at one point.

Abuse of Notation 58.4.5. We often don't mention x 0 and y 0 when they are understood (or irrelevant). For example, from now on we will just write S 1 ∨ S 1 for a figure eight.

Remark 58.4.6 -Annoyingly, in L A T E X \wedge gives ∧ instead of ∨ (which is \vee). So this really should be called the "vee product", but too late.

§58.5 CW complexes

Using this construction, we can start building some spaces. One common way to do so is using a so-called CW complex. Intuitively, a CW complex is built as follows:

• Start with a set of points X 0 .

• Define X 1 by taking some line segments (copies of D 1 ) and fusing the endpoints (copies of S 0 ) onto X 0 .

• Define X 2 by taking copies of D 2 (a disk) and welding its boundary (a copy of S 1 ) onto X 1 .

• Repeat inductively up until a finite stage n; we say X is n-dimensional.

The resulting space X is the CW-complex. The set X k is called the k-skeleton of X. Each D k is called a k-cell; it is customary to denote it by e k α where α is some index. We say that X is finite if only finitely many cells were used.

Abuse of Notation 58.5.1. Technically, most sources (like [Ha02] ) allow one to construct infinite-dimensional CW complexes. We will not encounter any such spaces in the Napkin.

Example 58.5.2 (D 2 with 2 + 2 + 1 and 1 + 1 + 1 cells) (a) First, we start with X 0 having two points e 0 a and e 0 b . Then, we join them with two 1-cells D 1 (green), call them e 1 c and e 1 d . The endpoints of each 1-cell (the copy of S 0 ) get identified with distinct points of X 0 ; hence X 1 ∼ = S 1 . Finally, we take a single 2-cell e 2 (yellow) and weld it in, with its boundary fitting into the copy of S 1 that we just drew. This gives the figure on the left.

(b) In fact, one can do this using just 1 + 1 + 1 = 3 cells. Start with X 0 having a single point e 0 . Then, use a single 1-cell e 1 , fusing its two endpoints into the single point of X 0 . Then, one can fit in a copy of S 1 as before, giving D 2 as on the right. Example 58.5.3 (S n as a CW complex) (a) One can obtain S n (for n ≥ 1) with just two cells. Namely, take a single point e 0 for X 0 , and to obtain S n take D n and weld its entire boundary into e 0 . We already saw this example in the beginning with n = 2, when we saw that the sphere S 2 was the result when we fuse the boundary of a disk D 2 together.

(b) Alternatively, one can do a "hemisphere" construction, by constructing S n inductively using two cells in each dimension. So S 0 consists of two points, then S 1 is obtained by joining these two points by two segments (1-cells), and S 2 is obtained by gluing two hemispheres (each a 2-cell) with S 1 as its equator.

Definition 58.5.4. Formally, for each k-cell e k α we want to add to X k , we take its boundary S k-1 α and weld it onto X k-1 via an attaching map

S k-1 α → X k-1 . Then X k = X k-1 ⨿ α e k α /∼

where ∼ identifies each boundary point of e k α with its image in X k-1 . §58.6 The torus, Klein bottle, RP n , CP n

We now present four of the most important examples of CW complexes. §58.

6.i The torus

The torus can be formed by taking a square and identifying the opposite edges in the same direction: if you walk off the right edge, you re-appear at the corresponding point in on the left edge. (Think Asteroids from Atari!)

Thus the torus is (R/Z) 2 ∼ = S 1 × S 1 .

Note that all four corners get identified together to a single point. One can realize the torus in 3-space by treating the square as a sheet of paper, taping together the left and right (red) edges to form a cylinder, then bending the cylinder and fusing the top and bottom (blue) edges to form the torus.

Image from [To] The torus can be realized as a CW complex with • A 2-skeleton with a single 2-cell e 2 , whose circumference is divided into four parts, and welded onto the 1-skeleton "via aba -1 b -1 ". This means: wrap a quarter of the circumference around e 1 a , then another quarter around e 1 b , then the third quarter around e 1 a but in the opposite direction, and the fourth quarter around e 1 b again in the opposite direction as before.

e 2

We say that aba -1 b -1 is the attaching word; this shorthand will be convenient later on. §58. 6 

.ii The Klein bottle

The Klein bottle is defined similarly to the torus, except one pair of edges is identified in the opposite manner, as shown.

Unlike the torus one cannot realize this in 3-space without self-intersecting. One can tape together the red edges as before to get a cylinder, but to then fuse the resulting blue circles in opposite directions is not possible in 3D. Nevertheless, we often draw a picture in 3-dimensional space in which we tacitly allow the cylinder to intersect itself.

Image from [In; Fr] Like the torus, the Klein bottle is realized as a CW complex with • One 0-cell,

• Two 1-cells e 1 a and e 1 b , and • A single 2-cell attached this time via the word abab -1 . §58. 6 

.iii Real projective space

Let's start with n = 2. The space RP 2 is obtained if we reverse both directions of the square from before, as shown.

However, once we do this the fact that the original polygon is a square is kind of irrelevant; we can combine a red and blue edge to get the single purple edge. Equivalently, one can think of this as a circle with half its circumference identified with the other half:

RP 2 RP 2

The resulting space should be familiar to those of you who do projective (Euclidean) geometry. Indeed, there are several possible geometric interpretations:

• One can think of RP 2 as the set of lines through the origin in R 3 , with each line being a point in RP 2 .

Of course, we can identify each line with a point on the unit sphere S 2 , except for the property that two antipodal points actually correspond to the same line, so that RP 2 can be almost thought of as "half a sphere". Flattening it gives the picture above.

• Imagine R 2 , except augmented with "points at infinity". This means that we add some points "infinitely far away", one for each possible direction of a line. Thus in RP 2 , any two lines indeed intersect (at a Euclidean point if they are not parallel, and at a point at infinity if they do).

This gives an interpretation of RP 2 , where the boundary represents the line at infinity through all of the points at infinity. Here we have used the fact that R 2 and interior of D 2 are homeomorphic.

Exercise 58.6.1. Observe that these formulations are equivalent by considering the plane z = 1 in R 3 , and intersecting each line in the first formulation with this plane.

We can also express RP 2 using coordinates: it is the set of triples (x : y : z) of real numbers not all zero up to scaling, meaning that (x : y : z) = (λx : λy : λz) for any λ ̸ = 0. Using the "lines through the origin in R 3 " interpretation makes it clear why this coordinate system gives the right space. The points at infinity are those with z = 0, and any point with z ̸ = 0 gives a Cartesian point since

(x : y : z) = x z : y z : 1

hence we can think of it as the Cartesian point ( x z , y z ). In this way we can actually define real-projective n-space, RP n for any n, as either (i) The set of lines through the origin in R n+1 , (ii) Using n + 1 coordinates as above, or (iii) As R n augmented with points at infinity, which themselves form a copy of RP n-1 . As a possibly helpful example, we give all three pictures of RP 1 .

Example 58.6.2 (Real projective 1-Space) RP 1 can be thought of as S 1 modulo the relation the antipodal points are identified. Projecting onto a tangent line, we see that we get a copy of R plus a single point at infinity, corresponding to the parallel line (drawn in cyan below).

S 1 0 0 1 0.36 R

Thus, the points of RP 1 have two forms:

• (x : 1), which we think of as x ∈ R (in dark red above), and

• (1 : 0), which we think of as 1/0 = ∞, corresponding to the cyan line above.

So, we can literally write

RP 1 = R ∪ {∞}.

Note that RP 1 is also the boundary of RP 2 . In fact, note also that topologically we have

RP 1 ∼ = S 1

since it is the "real line with endpoints fused together".

∞ 0 Since RP n is just "R n (or D n ) with RP n-1 as its boundary", we can construct RP n as a CW complex inductively. Note that RP n thus consists of one cell in each dimension.

Example 58.6.3 (RP n as a cell complex) (a) RP 0 is a single point.

(b) RP 1 ∼ = S 1 is a circle, which as a CW complex is a 0-cell plus a 1-cell.

(c) RP 2 can be formed by taking a 2-cell and wrapping its perimeter twice around a copy of RP 1 . §58.6.iv Complex projective space

The complex projective space CP n is defined like RP n with coordinates, i.e.

(z 0 :

z 1 : • • • : z n )

under scaling; this time z i are complex. As before, CP n can be thought of as C n augmented with some points at infinity (corresponding to CP n-1 ).

Example 58.6.4 (Complex projective space) (a) CP 0 is a single point.

(b) CP 1 is C plus a single point at infinity ("complex infinity" if you will). That means as before we can think of CP 1 as

CP 1 = C ∪ {∞}.

So, imagine taking the complex plane and then adding a single point to encompass the entire boundary. The result is just sphere S 2 .

Here is a picture of CP 1 with its coordinate system, the Riemann sphere.

Remark 58.6.5 (For Euclidean geometers) -You may recognize that while RP 2 is the setting for projective geometry, in
Ch 23-25 Quantum Ch 26-30 Calc Ch 31-34
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              An Infinitely Large Napkin

                Evan Chen

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An Infinitely Large Napkin
https://www.goodreads.com/book/show/42289622-an-infinitely-large-napkin
Cathy O'Neil8 books499 followers
2016-09-06T00:00:00Z
<div><div><div><article><section><section><span><a href="https://www.goodreads.com/review/show/1840054963">February 10, 2017</a></span></section><div><p><span>This was such a Malcolm Gladwell take on data science. I think this book touches on an important subject, and people should be aware of the issues O'Neil discusses. But instead of doing a deep dive into the subject, it just felt like a list of bad algorithms with instances of the people they hurt. It didn't contain many examples of "WMDs" that I had not already heard of, and that might be because she cited the New York Times for *like* all her case studies.</span></p><p>As someone who works in the field, I don't think this book was geared towards me. It wasn't technical or specific enough. Knowing her background, I really wanted her to get into the nitty gritty of some of the mathematics behind the algorithms. I know she is capable of doing this, but I think she instead chose to appeal to a wider audience. That's cool... except I don't think she did a great job of that either because this book lacked the context necessary to give people unfamiliar with the field a view of how machine learning and analytics typically work.</p><p>Ultimately, I thought the book felt unfocused, and it showed in the conclusion where proposed a series of pretty ridiculous recommendations within the span of about 15 pages. She has strong opinions on the topics she covered in the book. Although I agree with almost all of them, that isn't the point. The book is supposed to be about the algorithms, and instead she takes us on a tour of a collection of business and public policy malpractice, stating that the solution is to "encode values into our algorithms." Wut. Leaving the logistics of this aside... if the people doing the coding don't share your values (and they clearly don't), why would they do that? O'Neil herself noted a misalignment of incentives in many cases, particularly where the data work has been contracted out to other parties. Telling them to have values and trash their contracts is obviously not gonna fly! You can't fear-monger the whole book and then think everything will be happy sunshine rainbows once we take the data science Hippocratic oath. </p><p>I don't think there are easy answers, and I think it is ok to admit that. The takeaway should have been that data science isn't better than humans... because it is humans. We made the algorithms, so they run the gamut of use cases and demonstrate all the shades of gray we exhibit ourselves (yes, all fifty of them.)</p><p>TL;DR - Boo. I expected more from you, O'Neil.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/32466179?shelf=dfa-book-club"><span>dfa-book-club</span></a><span> </span><a href="https://www.goodreads.com/review/list/32466179?shelf=owned"><span>owned</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/30181442-yun"></a></section><section><span></span><span><p><span>562 reviews</span><span><span>28.7k followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/2159105101">February 28, 2022</a></span></section><div><p><span>I'm a data person. I pride myself on being logical and looking at the numbers before making decisions. And for quite a few years, I worked at a data visualization company and was a self-professed data geek. But can more data actually lead to worse results? That is what <i>Weapons of Math Destruction</i> tries to understand.</span></p><p>Insightful and timely, this book provides a detailed look at how algorithms based on big data don't always tell the truth or lead to a more fair world as they are purported to do. Rather, they contribute to a system that is opaque and hard to challenge, increasing the divide between the privileged and everyone else. </p><p>Each chapter provides a thoughtful exploration of an area where big data is supposed to be helping, such as college rankings, recidivism of convicts, applying for jobs, and getting loans. Algorithms in each area help define which are the best colleges, which convicts are most likely to reoffend, what personality types are best suited for a job, and who should get the best interest rates. That sounds useful, right?</p><p>But unfortunately, ideal data is not always available, so bad or irrelevant data is often used instead. And the resulting predictions are treated as gospel, increasing efficiency of the system, but harming those caught on the wrong side. It hurts a segment of the population while providing the rest of us with the false belief that fairness and justice has being done. In many cases, the algorithms' predictions create a negative feedback loop, directly influencing the outcome they were objectively trying to determine.</p><p>I found this book to be interesting and relevant. It really goes to show that your predictions are only as good as the data you've got. Whether you're a data geek like me or you just want to learn a little more about big data's potentially harmful effects, this is a worthwhile book to check out.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/30181442?shelf=favorites"><span>favorites</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/1826682-trish"></a></section><section><span></span><span><p><span>1,386 reviews</span><span><span>2,646 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1773837567">April 16, 2017</a></span></section><div><p><span>O’Neil deserves some credit right off the bat for not waiting until her retirement from the hedge fund where she worked to tell us the secrets of how corporations use big data (<i>our</i> data). Underlying the collection and use of big data is an attempt to utilize efficiencies in the market place for goods, money, and talent. Big data ostensibly can also “set us free” from time constraints and uneven knowledge dispersal. Conversely the opposite is often true. We are at the mercy of how our own data is shredded and packaged, and errors in the model can mean mutually assured destruction—for the school, corporation, family.</span></p><p>The book starts with examples any readers who actually picked up this book to read might recognize: the chances of getting into a major university. O’Neil doesn’t go into the actual algorithms but just explains the variables chosen to populate the algorithms. Just when I was wondering who this book is targeted at, since after all, we kind of know how to get into university already, she comes up with examples of big data messing with aspirations that are still (hopefully not) in our futures.</p><p>She addresses the real pain-in-the-ass nature of minimum wage jobs where the inadequate part-time hours are constantly changing to maximize profits for owners and to screw with employees ability to plan their life, their children’s lives, and the children’s caretaker’s lives. O’Neil addressed the situation in 2009 when Amex decided to reduce the risk of credit card nonpayment by reducing the credit ceilings on users who shopped at certain stores, like Walmart. She shows us the way micro-targeting ends up using data to perpetuate inequities in opportunity and “social capital.”</p><p>The hardest part of reading this book (there is no actual math), was keeping my mind on what O’Neil was saying. Every time she'd mention another example of the ways big data was screwing us over, my mind would wander to experiences of my own, or ones I’d heard from friends, family, or others. This is real stuff, and just when I thought that it would be an excellent book for those with skills and interest in social justice to take to an interview with Google, Amazon, or a big bank, in she comes with another example of how the “fixes” are almost worse than the disease (Facebook’s method of who your friends are determining your credit risk).</p><p>But O’Neil reminds us big data, mathematics, algorithms, etc. aren’t going to go away.</p><blockquote>"Data is not going away. Nor are computers—much less mathematics. Predictive models are, increasingly, the tools we will be relying on to run our institutions, deploy our resources, and manage our lives. But as I’ve tried to show throughout this book, these models are constructed not just from data but from the choices we make about which data to pay attention to—and which to leave out. Those choices are not just about logistics, profits, and efficiency. They are fundamentally moral."</blockquote>Exactly. We still have to use our brains, not just our computers. It is critical that we inject morality into the process or it will always be fundamentally unfair in some way or another, especially if the intent is to increase profits for one entity at the expense of another. One simply can’t include enough variables or specifics. Some universities have begun to audit the algorithms—like Princeton’s Transparency and Accountability Project—by masquerading as people of differing backgrounds and seeing what kind of treatment these individuals receive from online marketers. <p>O’Neil suggests that sometimes data might be used to good effect by targeting frustrated online commenters with solutions to their issues: i.e., affordable housing info, or by searching out possible areas of workplace or child abuse and targeting that area with resources. She wades into national election data and notes that only swing states get candidates attention, suggesting, by the way, that the electoral college has outlived its usefulness to the citizenry. Algorithms are not going to administer justice or democracy unless we find a way to use them as a tool to root out inequities and try to find ways to deliver needed services where they are deficient.</p><p>When I look at the totality of what O’Neil has discussed, I am inclined to think this book is best targeted to thoughtful high schoolers and college-aged students who are thinking about planning their careers, who have a penchant for mathematical and computer modeling, and who think their dream job might be with an online giant. I’d be happy to be disabused of this notion if someone wants to challenge my thought that much of this information is known to many of us who have been out of school for awhile and who have been paying attention to our online experiences and junk mail solicitations. But it is always interesting to read someone as coherent and on the side of social justice as Ms. O’Neil. </p><p>It might be noted that <a href="http://bit.ly/1iDRPlZ">Jaron Lanier in <i>Who Owns the Future?</i> (2013)</a> also talks about the use of big data to steer our thinking and makes a preliminary suggestion that individuals should be paid for their data—for data that is collected about them, for profit. It is an interesting discussion as well. Love these intersections of technology and humanity.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/1826682?shelf=business"><span>business</span></a><span> </span><a href="https://www.goodreads.com/review/list/1826682?shelf=computerology"><span>computerology</span></a><span> </span><a href="https://www.goodreads.com/review/list/1826682?shelf=economics"><span>economics</span></a></span><span></span> </ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/2118933-leo-walsh"></a></section><section><span></span><span><p><span><span><span>Author <i></i></span></span></span><span><span>3 books</span></span><span><span>123 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1841259889">April 5, 2017</a></span></section><div><p><span>Captivating. Insightful. And important. A 50,000 foot view of how automated big-data is a great tool for understanding human nature. How it has great promise to make our lives easy. And yet, a very real takedown of how systems engineers -- and corrupt power-seekers, like corporate executives and for-profit universities -- misuse this powerful tool. And the even worse cases where people start with good intentions, like ridding school systems of bad teachers, only to toss out "false negatives."</span></p><p>I found Cathy O'Neil's "Weapons of Math Destruction" a very important book that highlights a lot of what's been going on in America over the past 30 or 40 years. For instance, I just finished "The New Jim Crow," and wondered how the Supreme Court would continue to rule in favor of crime policing tactics that 1) target poor urban areas -- populated by mostly black men, and 2) allow the police the ability to stop people, willy-nilly, that they found "suspicious" -- despite the fact that minorities are caught in stop-and-frisk situations while most are innocently going about their lives. Problem is, many of these people going to jail for nuisance crimes -- possessing small amounts of marijuana, open containers, driving on expired tags, etc. Things that seem the exact definition of "systematic racism." </p><p>But when O'Neil lays out how systems engineers have written algorithms that send police to "hot spots," those rulings make sense. For instance, large crimes that we all want to stop -- car thefts, burglaries, rapes, assault and murder -- are rare. While petty and nuisance crimes -- jaywalking, possessing weed, noise violations, vandalism, etc. -- common. Based on the law of large numbers, a program trying to optimize "broken windows" policing would send more officers to a district with a higher concentration of people, typically ghettos. Which leads to more petty arrests in those areas for crap that doesn't matter much. Which sends even more police into these areas, making more arrests for petty offenses... and so on and so on. </p><p>In short, these ghettos are caught in a negative feedback loop. And the residents more likely to end up in jail for something stupid. Like smoking dope which, based on most evidence, is just as high among whites than blacks. So a white 19-year-old frat boy at Ohio State can smoke-up at will, with little possibility of being caught. While a nearby ghetto-dweller, who works maintenance or in the office at the university, or attends the university while living at home, will have a greater likelihood of being arrested. Just based on where they live.</p><p>Same crime -- use of narcotics. Two different outcomes. </p><p>That is what O'Neill terms as a "weapon of math destruction." Since it is pervasive, destructive, and opaque. </p><p>And things get worse if the two hypothetical dope-smokers get arrested. Odds are, the white frat boy's parents live in a safe suburban neighborhood and thus knows zero convicted felons. So his court-appointed "recidivism" score -- attained by another destructive math weapon -- will be lower than the ghetto dweller's who lives near many felons. Thus, the courts may lessen the frat boys charge to a misdemeanor while charging the ghetto dweller with a felony. Due to that recidivism score. </p><p>Talk about kicking someone when they're down.</p><p>And when both released on parole, and ordered to "steer clear of felons." the frat boy will have no problems following this. While the ghetto kid, thanks in part to the policing software noted above, will be surrounded by them. Which, of course, adds to the already disadvantaged the further risk of being pegged as a parole violator. </p><p>Smack. </p><p>O'Neill lays out other ways that system engineers perpetuate injustice. She takes-down for-profit universities -- like the University of Phoenix -- who actively target poor people with aspirations. Not to help, since a University of Phoenix degree costs tens of times more than a community college degree, while adding less salary. Instead, University of Phoenix and their ilk exist to cash in the student loans guaranteed by the government. </p><p>Yep. Poor people, who don't know any better, are targeted by companies that give them less while charging them more. Using data-driven web advertising, another Weapon of Math Destruction, all for a quick buck. </p><p>These are just a few WMD's that O'Neill examines. She looks at others credit scoring, e-scoring (a sort of electronic "credit score" derived on you based on your social media friends and activities), the USA Today college rankings, which have lead to spiraling tuition costs while providing questionable value. </p><p>All of these WMD's lead to increasing social stratification. And, in many ways, drive the "winner take all" nonsense that gives big money to a handful of developers who program an app.</p><p>But the nice thing is that O'Neill ends by providing valuable insight into how, when properly used, deep-dive statistics can actually help people. For instance, O'Neill was part of a task-force that examined New York City's homeless. They uncovered the single unequivocal variable that would keep people off the streets -- access to Section 8 housing. And once these people were housed, they'd move on to get jobs --since having a stable residence makes it easier to gain employment. And thus, less likely to end up on the streets. And all this research came at a time when Mayor Bloomberg was contemplating reducing Section 8 housing. So it's prove important.</p><p>She also points to other positive uses of algorithms -- all of which point to putting our compassionate, human-based morality ahead of the appearance of objective, measurable efficiency -- with "appearance" being the operative word. Since O'Neill makes the cost of following these algorithms clear. </p><p>All-in-all, "Weapons of Math Destruction" is the best science book I've read over 2016. Since it focuses not only what we can do -- the science and how it makes things more efficient -- but forces us to focus on the ethics, the "why" we may choose a less efficient alternative as it may be more just. Especially when blindly accepting a model often degenerate to "pseudoscience." And that anti-scientific narrative can be amplified if the person wielding the WMD is either greed or malicious. </p><p>5-stars.</p><p>That said... YAY!!! This is my 80th book of the year. So I've just completed my goal.</p><p></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/793473-clif-hostetler"></a></section><section><span></span><span><p><span>1,173 reviews</span><span><span>878 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1784597657">January 18, 2019</a></span></section><div><p><span>"Welcome to the dark side of big data." Thus the author concludes the Introduction section of this book. Computers and the internet have enabled us to advance into the new world of algorithms and big data with ramifications that most people are unaware of. </span></p><p>Surfing the web, clicking "like" in Facebook, Googling (i.e.searching on line), and making online purchases are common examples where big data is either tracking and potentially impacting our lives. Some of these are benign and can be helpful. But this book focuses sharply in the other direction, on the "harmful examples that effect people at critical life moments, going to college, borrowing money, getting sentenced to prison, finding and holding a job. All these life domains are increasingly controlled by secret models wielding arbitrary punishments." </p><p>One of the most shocking pieces of information provided by this book are the examples of how big data contributes to economic inequality. It is devastating how efficiently private colleges and payday loan companies target the economically stressed portion of the population. It's also astounding how frequently limitations and flaws of big data are ignored and substituted for the truth. It's almost comical how big data systems can be manipulated (a.k.a gaming the system) by clever institutions and companies. </p><p>Whether we like it or not this is the new world in which we all live. Citizens of this new environment need to become knowledgeable about how big data works. Otherwise we will all be its clueless victims. </p><p>The following are some quotations from the book.</p><p>The following are the closing sentences of the Introduction:</p><blockquote>Big data has plenty of evangelists, but I'm not one of them. This book will focus sharply in the other direction, on the damage inflicted by WMDs and the injustice they perpetuate. We will explore harmful examples that effect people at critical life moments, going to college, borrowing money, getting sentenced to prison, finding and holding a job. All these life domains are increasingly controlled by secret models wielding arbitrary punishments. Welcome to the dark side of big data.</blockquote>In the following the author defines some of the shortcomings she observed in how big data was being used.<blockquote>More and more I was worried about the separation between technical models and real people and about the moral repercussions of that separation. In fact I saw the same pattern emerging that I had witnessed in finance, a false sense of security was leading to widespread use of imperfect models, self-serving definitions of success, and growing feedback loops. Those who objected were regarded as nostalgic Luddites. </blockquote>The author worked at a hedge fund during the 2008 financial collapse. Thus when she moved into the field of consumer data modeling she looked for flaws in the use of data that were similar to what led to the credit crisis.<blockquote> I wondered what the analog to the credit crisis might be in big data. Instead of a bust I saw a growing dystopia with inequality rising. The algorithms would make sure that those deemed losers would remain that way. A lucky minority would gain evermore control over the data economy raking in outrageous fortunes and convincing themselves all the while that they deserved it. </blockquote> The following is the author's summary near the end of the book.<blockquote> In this march through a virtual lifetime we've visited school and college, courts and the work place, even the voting booth. Along the way we have witnessed the destruction caused by WMDs. Promising efficiency and fairness, they distort higher education, drive up debt, spur mass incarceration, pummel the poor at nearly every juncture, and undermine democracy. </blockquote>The author, Cathy O'Neil is highly qualified (she got her Ph.D. in math at Harvard), had work experience from inside the system (a “quant” at D.E. Shaw—a major hedge fund), and has evolved into a "Occupy Wall Street" activist. Thus she has experience and qualifications needed to explain how big data effects our lives. It's needed information.<p>Author's Ted Talk:<br/><a href="https://embed.ted.com/talks/cathy_o_neil_the_era_of_blind_faith_in_big_data_must_end">https://embed.ted.com/talks/cathy_o_n...</a></p><p>Cathy O'Neil was the winner of the 2019 MAA Eurler Book Prize:<br/><a href="https://www.maa.org/programs-and-communities/member-communities/maa-awards/writing-awards/euler-book-prize">https://www.maa.org/programs-and-comm...</a></p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/793473?shelf=current-events"><span>current-events</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/175635-trevor"></a></section><section><span></span><span><p><span>1,362 reviews</span><span><span>23.1k followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/2507802271">August 26, 2018</a></span></section><div><p><span>We like to think of mathematics as basically pure and free from the nastier side-effects of human nature. And this purity rubs off – so that the closer a science is to being able to be described in numbers, the more highly we regard it – so, physic is seen as somehow higher than biology, and economics than anthropology. That is, if you can predict behaviour on the basis of an algorithm – whether that be the behaviour of a billiard ball or a homeless person – then this is proper science and it has a claim to a kind of objective truth that sets it aside from being challenged.</span></p><p>And that is the point of this book – it seeks to help shatter this illusion, particularly in relation to the human sciences, but also, and perhaps more importantly, in relation to marketing, insurance, policing, education and other social activities that are increasingly being modelled and even normalised by algorithms. She refers to these algorithms as the weapons of math destruction in the title (the title is better in English, of course, where we say maths, rather than math, but her point stands). </p><p>The destruction such algorithms can cause became clear to her while she was working in finance just before the 2008 crash. It certainly isn’t that she sees maths itself as being the problem, she refers to herself almost immediately as a kind of math nerd and proud of that designation. Her point is that these algorithms are dangerous because we tend to think of them as being purely objective and therefore the results they provide as being beyond question. And this state is helped along by more than just the fact we hold mathematics in such high regard. It is furthered by the fact that often the algorithms that spit out these assessments of us are obscure, unfair and grow exponentially. These are the three conditions that the author believes makes an algorithm a likely WMD. </p><p>So, in looking at these in turn, is the algorithm is obscure? – and most of them tend to be, as she says at one point, they are the ‘special sauce’ for many companies and so they need to remain hidden from the competition. For instance, if you have an algorithm that allows you to predict who is going to make an ideal partner for someone else, then your dating site is going to make you lots of money. You are hardly going to want to let your competition know you secret. But the problem is that by keeping your algorithm obscure and hidden from outside analysis, you can say nearly anything you like about its effectiveness, if no one is then able to check. You know, this is a version of the ‘they can vote any way they like, as long as I get to count’ story.</p><p>But it isn’t just outright fraud that is the problem here – although, that doesn’t mean fraud isn’t a problem. Just as bad is the idea that often these models bury their false negatives. That is, if the algorithm says ‘never employ anyone with green eyes, they are under-performers’ the company that follows this advice is unlikely to ever find out if this is true or not. That’s because they won’t have employed anyone with green eyes to test the model – so the model will be confirmed by default. The problem is that many of the algorithms used – say in policing – can encourage over-policing of certain populations (this is set in the US, so let’s just call those certain black, Hispanic and Muslim populations to save time) and this over-policing, by defining certain populations in these ways are also likely to create the monster they were supposed to be eliminating. A nice example is an algorithm that denies jobs to people according to their credit scores, which then means these people are less able to pay off their debts, which gives them a worse credit score, which then confirms the risk – vicious cycle anyone?</p><p>Other examples focus on the use of psychometric tests for all manner of things, but increasingly as a pre-employment test. You might think these ought to be relatively easy to game – you know, how stupid would you have to be to answer ‘very true’ to the prompt, ‘I sometimes fly off the handle for no real reason’. You know, unless you are going for a job in as a wrestler, your employer is probably not going to be overly impressed with that answer. But as she points out, sometimes applicants, for example, for a job at McDonalds, are asked to choose between one of two alternatives: “It is difficult to be cheerful when there are many problems to take care of – or – Sometimes, I need a push to get started on my work”. I have no idea what the ‘right’ answer to that choice is. I’m not even sure which of the two really applies to me – although, there have been times in my life when both of them have, and in your life too, I suspect. That is, it isn’t at all clear what this is seeking to achieve, but it is clear that there is likely to be a ‘wrong’ answer, in the sense that making that ‘choice’ might leave you without a job. </p><p>But even this level of obscurity isn’t her main worry (at least you know you’ve been asked a question here that might be used against you) – but it is too often easy to correlate factors to ensure that certain people are excluded due to their sex, sexuality, race, social class and so on, merely by the underlying assumptions of those programming these algorithms – and the algorithms aren’t ‘objective’ examples of the purity of mathematics, but rather socially produced ephemera that are likely to have been shaped by the social stereotypes of the society and the people who produce them. </p><p>The author mentions that curious fact that orchestras now appoint five times as many female players since auditions have been held with the player behind a curtain. Who’d have thought women would play so much better when they were hidden from sight? Such shy and retiring little things, bless them… </p><p>The solution is to ensure that algorithms used to judge us are open and available for anyone to check – a condition that will become increasingly unlikely as these algorithms are proprietary.</p><p>The question of fairness isn’t at all easy to address and for many of the reasons already mentioned. One of the examples given is teacher scores being used to determine who should get pay increases and who should be removed from the profession. Basically, the idea is to compare student outcomes so that the teachers who do not increase student scores enough should be shown the door. But, as is pointed out here, student scores aren’t only affected by teacher performance. And worse, if you are teaching students who are either very far behind or very far in front you are unlikely to affect as big a movement in their scores as you might if you are teaching kids in the middle. If you are going to be assessed on the basis of a score, that score ought to be related to something you have control over, rather than merely something that is relatively easy to measure. The author points out that far too often the kinds of assessments made of teachers based on student attainment produce virtually random results year on year. Since it is very unlikely that an exceptional teacher will become a very poor teacher from one year to the next, then any assessment that produces such a result ought to be suspect. And this is also true of loan application algorithms that judge you based upon the people you associate with. For instance, I’ve read a few times lately that your credit rating can be impacted by your ‘friends’ on FaceBook, but even if this is not literally true, algorithms are shown here to judge you by many factors that assume associations between you and other people more or less likely to be a credit risk. For instance, if you buy things in certain stores, as American Express admitted recently they restricted access to credit to people based on them frequenting particular stores. </p><p>The last condition of a WMD is that the algorithm can be scaled up to cover large populations. An algorithm that works well in one set of circumstances might be both transparent and fair within its limited application, but because it is then being used across a larger population it might suddenly stop being fair or reasonable. This is because it creates a new norm in the population at large and that might well have seriously disadvantageous impacts on populations beyond the one it was originally intended to be used upon. Again, the examples I jump to tend to be associated with education, where large scale testing programs have a disproportionate impact on poor communities which are defined as failing and then ‘success’ becomes defined as ‘doing well on the test’ – so that the tail starts wagging the dog. And this then has impacts on how kids get taught – if you are only going to be measured by test results, then we should just prime you for taking tests. Which then makes education as boring as it is possible to make it for kids that were already struggling to see the point of education in the first place. But the communities that do well on these tests are normally the already advantaged and they suffer none of these negative impacts, because, well, they do well on these tests, so no point priming them on more of them…Let them do art.</p><p>I really would recommend this book – she gives lots and lots of examples and it is vitally important that we understand that this is the world we are increasingly moving towards. More and more of our lives are going to be influenced by algorithms and big data – and yet too many of us are so terrified of mathematics that we will blame ourselves when these algorithms punish us. It isn’t at all clear to me how we might go about making these algorithms transparent, fair or limited to a scale that keeps them safe – but these are questions we really ought to think about and act upon.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/175635?shelf=economics"><span>economics</span></a><span> </span><a href="https://www.goodreads.com/review/list/175635?shelf=education"><span>education</span></a><span> </span><a href="https://www.goodreads.com/review/list/175635?shelf=maths"><span>maths</span></a></span><span></span> </ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/5917740-nilesh-jasani"></a></section><section><span></span><span><p><span>1,083 reviews</span><span><span>204 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1822772361">November 27, 2016</a></span></section><div><p><span>For the most part, WMD is a rant with only impractical statements as solutions. </span></p><p>The author is onto something critically important when one reads the title of the book and goes through the first few pages. However, it is a tragedy to see the author falling in love with her own phrase WMD and completely losing the plot. The examples used are good in the beginning but soon turn ridiculous (they would be laughable if not so lamentable for the people involved). In the process, the author loses her credibility as a champion of the topic.</p><p>Consider the following example: a person is unable to find a job as he claims he fails all pre-interview personality tests. The author debunks the objectivity of tests in favor of the subjectivity of interviewers with the claim that in another world, one or the other interviewer would have given the job to this person. The author never thinks about the interviewers' non-transparent and more difficult to fathom biases wreaking greater damage in that world versus the current one where the biases are at worst in models that are far easier to scrutinize and amend. The author failts to realize that every world with fewer desirable jobs compared to applicants will leave some unhappy - the fairness is perhaps far better where the selection process is more equal for all.</p><p>In a way, the author wishes "we" to stand up and help reduce the use of math. The author is clear in the "reduction" goal without ever realizing how "we" would ever decide to what extent the reduction should take place and whether the constituents of "we" could ever agree on the replacement. The repeated call for subjective biases assumes that the biases come in only one form and are acceptable to all. The absurdities reach a peak when the author declare that at least subjective biases evolve over time and hence should be preferred while the mathematical models are static!</p><p>The author's hatred for anything involving models not only creates a highly one-sided discourse but also leads to inadequate treatment of any important points within. The only mathematical concept discussed somewhat is that of the circularity caused by the self-reinforcing feedback in multi-equilibrium equations leading to solutions that might be locally optimum but not globally (all my words). Otherwise, the book almost exclusively focuses on the unsuitability of results arrived through statistical sampling on any individual case.</p><p>To summarize, the author seems to yearn for insurance companies to abandon actuarial methods, schools to move away from test scores for admissions and financial institutions/tech companies/governments to reduce their use of numbers. The author wants humans to take decisions that these models might be taking using their value judgments. Or the world to simply not take those decisions. </p><p>There is so much sensible that needs to be discussed on the damage created by the excessive use of math in the modern world but this book is not the best place to explore it.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/5917740?shelf=non-fiction"><span>non-fiction</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/78485297-mario-the-lone-bookwolf"></a></section><section><span></span><span><p><span>805 reviews</span><span><span>4,960 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/2347067629">December 11, 2018</a></span></section><div><p><span>Is it legitimate to reduce people to the data that can be extracted from them?</span></p><p>Please note that I put the original German text at the end of this review. Just if you might be interested.</p><p>Especially the predictions made possible by big data are a frightening aspect so that behavior and personal development can be predicted with increasing probability. Also, like any artificial intelligence, the algorithms and programs become more and more efficient, both in parallel with the growing amount of data about the individual and through the optimization of the functioning. <br/>Whether habits, buying behavior, Facebook Likes, movement patterns, writing style, illnesses, search machine inputs, political activities, professional aspects or social contacts, everything flows together in a large memory.</p><p>Particularly explosive is the ability to accurately predict the electoral behavior of specific groups of the population and thereby be able to address election advertising even more targeted. It also opens up the option of manipulating groups, which are more likely to be responsible for the competition camp, utilizing disinformation and misinformation in their awareness-raising and decision-making. Even outside of elections, the data volumes offer promising perspectives for influencing opinions, facts, trends and mood barometers.</p><p>The subtle, digital prompters are hardly recognized as such. The information is automatically trimmed in the form of auto-completion and suggestion functions, the individually compiled search results and purchase recommendations based on the previous buying behavior. Moreover, directed to the individual interests or the opinion to be shaped. Even what you type in and then delete again or also revise and change, is saved.</p><p>This can be considered optimistic as practical, time-saving and relieve and helper at work. Alternatively, as an indirect kind of influencing, over which only the compulsory consumption is promoted as optimized as possible. As an alternative to the offer of tailor-made products, there is the option of deliberately focusing just on specific information and news. So that unpopular and annoying reports, opinions and activists are intentionally not even displayed in the higher result ranges. To merely skip the scissors in the head and defamation immediately and proceed directly to censorship.</p><p>Perfide is the playful way in which free offers lure the consumer. For example, a health application for smartphones, with which one competes in real time with friends, in which insurance and health insurance should also be interested in. These hands-on free programs provide a fun and competitive and user-driven option for businesses to access users' location, interest, and behavioral data. </p><p>The book is an excellent appetizer, with the benefit of awakened interest in possible deepening of the study of the topic by further and detailed, related literature. <br/>Given the overall situation a worthwhile endeavor. Contrary to the apathy and disinterest of large populations owed the fact that it could come too far. So without long-awaited outcry across all layers of civil society, much more could go on. </p><p>In Western countries, social networks, search engines, and Internet merchants are only interested in manipulating buying behavior and collecting data to monetize it. The direction of China with the Citizen Score is likely to find imitators. Then, the arbitrary and negligently disclosed data would not anymore only decide the price of a product or the next banner ad. However, about your future and maybe your survival or death. For example, if you criticize the government.</p><p>Ist es legitim, Menschen auf die Daten zu reduzieren, die sich aus ihnen extrahieren lassen?</p><p>Gerade die durch Big Data möglich werdenden Prognosen stellen einen erschreckenden Aspekt dar, lässt sich Verhalten und persönliche Entwicklung auf diese Weise doch mit immer höherer Wahrscheinlichkeit voraussagen. Und wie jede künstliche Intelligenz werden die Algorithmen und Programme sowohl parallel zu den wachsenden Datenmengen über die jeweilige Person als auch durch die Optimierung der Funktionsweise immer effizienter. Ob Gewohnheiten, Kaufverhalten, Facebook Likes, Bewegungsmuster, Schreibstil, Krankheiten, Suchmaschineneingaben, politische Aktivitäten, berufliche Aspekte oder Sozialkontakte, alles fließt in einem großen Speicher zusammen.</p><p>Besonders brisant ist die Möglichkeit, Wahlverhalten bestimmter Bevölkerungsgruppen präzise vorhersagen zu können und dadurch Wahlwerbung noch zielgerichteter adressieren zu können. Es öffnet auch die Option, eher dem Konkurrenzlager anzurechnende Gruppen gezielt mittels Des- und Falschinformation in ihrer Bewusstseinsbildung und Entscheidungsfindung zu manipulieren. Auch abseits von Wahlen bieten die Datenmengen verheißungsvolle Perspektiven für die Beeinflussung von Meinungen, Fakten, Trends und Stimmungsbarometern.</p><p>Dabei werden die subtilen, digitalen Souffleure nur schwerlich als solche erkannt. In Form von Autovervöllständigungs- und Vorschlagfunktionen, den individuell zusammengestellten Suchergebnissen und auf das bisherige Kaufverhalten angelehnte Kaufempfehlungen werden die Informationen automatisch beschnitten. Und auf die individuellen Interessen oder die zu prägende Meinung hin gelenkt. Selbst was man eintippt und dann wieder löscht oder noch überarbeitet und verändert, wird gespeichert. <br/>Das kann optimistisch als praktisch, zeitsparend und die Arbeit erleichternd betrachtet werden. Oder als eine indirekte Art der Beeinflussung, über die vorerst nur der Konsumzwang möglichst optimiert gefördert wird. Alternativ zur Offerierung der auf den Leib geschneiderten Produkte bietet sich die Option, gezielt nur bestimmte Informationen und Nachrichten in den Fokus kommen zu lassen. Damit missliebige und lästige Berichte, Meinungen und Aktivisten gezielt gar nicht erst bis in den höheren Ergebnisrängen angezeigt werden. Sondern die Schere im Kopf und Diffamierung gleich zu überspringen und direkt zur Zensur voranzuschreiten.</p><p>Perfide ist die spielerische Art und Weise, mit der der Konsument durch Gratisangebote geködert wird. Zum Beispiel eine Gesundheitsapplikation für Smartphones, mit der man in Echtzeit mit Freunden konkurriert, was Versicherungen und Krankenkassen auch interessieren dürfte. Diese praktischen kostenlosen Programme ermöglichen eine spielerische und auf Wettbewerb und den Ehrgeiz der Anwender zielende Option für Unternehmen, an Standort-, Interessens- und Verhaltensdaten der Benutzer zu kommen.</p><p>Das Buch ist ein guter Appetitmacher, mit dem Interesse an einer etwaigen Vertiefung der Beschäftigung mit dem Thema durch weiterführende und auf einzelne Aspekte detaillierte eingehende Literatur geweckt wird. Angesichts der Gesamtsituation ein erstrebenswertes Unterfangen. Entgegen der, der Apathie und Desinteresse breiter Bevölkerungsgruppen geschuldeten Tatsache, dass es so weit kommen konnte.Und, ohne längst fälligen quer durch alle Schichten der Zivilgesellschaft gehenden Aufschrei, noch viel weiter gehen könnte.</p><p>In den westlichen Ländern sind die sozialen Netzwerke, Suchmaschinen und Internethändler nur an der Manipulation des Kaufverhaltens und dem Sammeln von Daten zu dessen Monetarisierung interessiert. Die Richtung, die etwa China mit dem Citizen Score geht, dürfte Nachahmer finden. Dann entscheiden die willkürlich und fahrlässig preisgegebenen Daten nicht nur mehr über den Preis eines Produkts oder das nächste Werbebanner. Sondern über die eigene Zukunft und vielleicht auch das eigene Überleben oder Sterben. Wenn man etwa die Regierung kritisiert.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/78485297?shelf=0-internet"><span>0-internet</span></a><span> </span><a href="https://www.goodreads.com/review/list/78485297?shelf=0-social-criticism"><span>0-social-criticism</span></a><span> </span><a href="https://www.goodreads.com/review/list/78485297?shelf=0-technology"><span>0-technology</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/5096743-suzanne"></a></section><section><span></span><span><p><span>1,726 reviews</span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1735584635">September 7, 2016</a></span></section><div><p><span>This book did a nice job describing large-scale data modeling and its pitfalls in a very accessible manner. It is so easy to think of computer algorithms as unbiased; however, the author demonstrates how they really do discriminate. Next time I teach a class involving statistics, I may use this book to show students how it is dangerous to blindly believe the numbers.</span></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/5096743?shelf=netgalley"><span>netgalley</span></a><span> </span><a href="https://www.goodreads.com/review/list/5096743?shelf=nonfiction"><span>nonfiction</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/155663-david-rubenstein"></a></section><section><span></span><span><p><span>828 reviews</span><span><span>2,690 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/2210106771">December 10, 2017</a></span></section><div><p><span>The subtitle of this book, <i>How Big Data Increases Inequality and Threatens Democracy</i> really says it all. Big data has come into our lives in numerous ways, and many of them are a scourge on our lives. Big data, in and of itself, is not to blame, but the uses to which it is put are often outrageous. Take the case of automated teacher evaluations. These are often based on the improvement of students' scores. It seems like a no-brainer, and since the scores take into account the <i>improvement</i> rather than the absolute scores, they seem to be very fair. However, one New York teacher received an abysmal score of 6 (out of 100) one year, and the following year received a wonderful score of 96. Obviously the teacher did not suddenly improve his teaching methods.</span></p><p>If a mathematician were to analyze the scores from such evaluations, their random scatter would be instantly recognized as meaningless. Yet, these automated evaluations are used for hiring/firing decisions, as well as compensation decisions. The worst aspect, is that there is no attempt to improve the algorithms. The algorithms used in these evaluations are opaque, and no attempt is made to apply feedback to tweak the scores to make them more accurate and fair.</p><p>Such algorithms are used in many avenues of life. Credit scores are used by loan companies--and even by auto insurance companies! Police departments use algorithms to plan targeting of police activity, while courts use algorithms to predict recidivism. Colleges, especially for-profit colleges, use algorithms to target potential students.</p><p>While this book comes across as very preachy, and very liberal, it does make some very good points. Big data is often used--intentionally or not--to punish the down-trodden and to increase inequality. Profitability is usually the goal, and while fairness is often the purported goal, these algorithms rarely turn out to have the effect of being fair.</p><p>I recommend this book to people who are curious about the effects of big data on their lives and on democracy.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/155663?shelf=mathematics"><span>mathematics</span></a><span> </span><a href="https://www.goodreads.com/review/list/155663?shelf=politics"><span>politics</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/2431873-emily"></a></section><section><span></span><span><p><span>735 reviews</span><span><span>2,444 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1809334375">February 7, 2017</a></span></section><div><p><span>Big Data is opaque, complicated, managed by profit-seeking corporations, and is more and more dictating certain societal conditions: from getting a job to applying to college to receiving healthcare. "Data," on its own, seems amoral, a way to implement systems that are more fair. But O'Neil's point in this book is that all algorithms include basic assumptions, and sometimes those basic assumptions are full of bias and not grounded in fact. If the algorithms aren't regularly inspected, they create their own feedback loops. As O'Neil says, "In each case, we must ask not only who designed the model but also what that person or company is trying to accomplish."</span></p><p>This book touches lightly on a number of topics to give an overview of the algorithms that O'Neil finds the most objectionable. Each of the different sections is interesting, but they don't delve particularly deeply into the topic; for example, if you're interested in how poor feedback loops on policing and prisoner recidivism work together to create our prison industrial complex, you're better off reading <a href="https://www.goodreads.com/book/show/6792458-the-new-jim-crow">The New Jim Crow</a>. But if you're interested in an introduction to the ways in which data runs our lives, this is a good place to start. </p><p>I was particularly interested in the section on college admissions, because O'Neil ties skyrocketing college tuition prices to the US World News ranking (the first ranking for universities in the US). In 1988, journalists used a number of proxies to create a way to rank colleges against each other. This example hits on a few of the points that O'Neil brings up again and again: the journalists wanted proxies that would reaffirm their own biases, so Stanford and Harvard had to be at the top of the list. But they weren't able to fully factor in the things that mattered, like the quality of education, so they relied on proxies like admissions rate and freshman retention rate. Cost was conspicuously excluded, and thus - or so O'Neil says - college tuition has increased 500 percent between 1985 and 2013. Colleges are gaming the system to get themselves back up to the top of the list, but then are passing those costs directly on to the students who attend.</p><p>While this book is nothing groundbreaking, it's another reminder that tech HAS to do better if we are going to put more of our personal data - our lives - into its hands. I personally believe we're all being swindled by the "sharing economy" (heavy quotes there), and the example about Lending Club hammers that home. O'Neil uses Lending Club as an example of a service that was supposed to "democratize" loans, but it's quickly devolved into 80% institutional money: that is, money from big banks. Why? Banking has regulations, and banks aren't legally allowed to discriminate against consumers on the basis of e-scores and other algorithmically created scoring systems that take into account factors like zip code, punctuation on applications, and social networks. These institutions have found a way to get outside of the law, apply their sketchily drawn "data" models (that have no statistical basis), and reap the rewards, while further exacerbating class inequality.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/2431873?shelf=politics"><span>politics</span></a><span> </span><a href="https://www.goodreads.com/review/list/2431873?shelf=science"><span>science</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/30018093-scott"></a></section><section><span></span><span><p><span>34 reviews</span><span><span>4 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1793743506">October 25, 2016</a></span></section><div><p><span>Book reviews are all about expectations, and honestly I, as someone doing data science and grappling with issues, expected more. With a data scientist writing a full length book inditing data science one expects a deep dive revealing real points. Instead it ends up being a very surface level essay without the deeper exploration and meaning one expects from a full length work. Perhaps more worrisomely, her own definition of a WMD it introduces is often worked around to bring in arguments she wants to make.</span></p><p>I do agree with her main argument/definition of a Weapon of Math Destruction(WMD), though the name is far too punny for real use. More or less the author argues that bad algorithms are secret, and expand out rapidly without checks on actual results. This is hardly a novel problem statement, but it is certainly a real problem. However several times she applies shortcuts of assumed results to actual results in evaluating presumed WMDs which is the very same problem she is complaining about. She also seems to have added an unspoken fourth criteria being anything that offends her liberal (and she is very liberal, and interventionist) ideals. This fourth criteria is apparently enough to give a pass to what are clearly bad algorithms that fail her own tests. This is needless to say, not a great way to prevent issues with unethical use of algorithms, unless you happen to have the same ideals and goals as Cathy O'Neil. </p><p>Somewhat surprisingly, this is a data science book without any data. It doesn't even have in text footnote (at least in my kindle edition, though they do sometimes update that), and the notes section at the end shows the citations to be really quite poor and sparse. We are talking college freshman "The professor said 10 citations so I better cite a bunch of random webpages" level of citations. To be fair her arguments are only weakly data driven, but I still expected something with a deeper academic backbone.</p><p>This is also a data science book that won't teach you anything about data science. If you are looking for that you should look elsewhere. The author has co-written another book <a href="https://goodreads.com/book/show/17346997.Doing_Data_Science">Doing Data Science</a>, and it shows her quite capable of handling the topic. I do wish she had brought more of that into this book, which will be many people's first book in data science. </p><p>The structure of the book plays out more as a series of related blog posts than a book that builds on themes and has progression. Partly this may be an unfortunate result of the author being a blogger, but it leaves one wanting more. </p><p></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/6816078-maru-kun"></a></section><section><span></span><span><p><span>220 reviews</span><span><span>522 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/2094214046">August 22, 2017</a></span></section><div><p><span>Forget those cute pastel illustrations from the fifties with their flying cars, robot servants and dreams of unlimited leisure. Our future has finally arrived and for most of us, especially for the less rich and less privileged who won’t qualify for individualized attention, the computer says “No”.</span></p><p>‘Weapons of Math Destruction’ is a timely book about the increasing influence of algorithms to control the news we see, the jobs we can get and the politicians we vote for; algorithms working tirelessly on someone’s behalf (not yours), unseen and unaccountable.</p><p>The book explains statistical and methodological problems with these algorithms and illustrates how these same problems manifest themselves when they are applied to real world situations.</p><p>An example gives a flavor of the issues that recur throughout the book:</p><p>An algorithm which compared changes in student performance year-on-year was used to decide which teachers were a poor performers who would be let go. In one terminated teacher’s case subsequent evidence suggested that student’s scores had been tampered with the previous year and as a result she had inherited a class whose performance had been overstated, so was bound to deteriorate with the algorithm marking her out as a poor teacher.</p><p>The teacher was not told how the algorithm applied nor allowed to appeal the decision; the algorithm was a black box that produced a result the school system wanted – terminations. And, notwithstanding the lack of transparency in this decision, a sample of scores from one class alone - perhaps twenty to thirty students - was not sufficient to produce a statistically valid result in any event. One more reason why being an American public school teacher must be one of the worst teaching jobs in the developed world.</p><p>Please read the book for more egregious examples of “algorithm abuse”. Below is my summary are some of the key problems the book outlines. It is helpful to think about these problems in two categories, problems which come from poor application of the technical aspects of the algorithms (bad programming, misapplication of statistical or machine learning methods) and problems which come from how the algorithms are developed and used, how they incorporate hidden biases or value judgments, poorly thought out objectives and other ‘human factors’ beyond just poor maths:</p><p>Some of the common issues were:</p><p>	Lack of transparency in how the models operate and how they make a decision, often leading to no means of appeal against a clearly unjust outcome. Related problems include over reliance on models in the face of contradictory data or, where people do understand the models but are benefitting from them, a lack of integrity in applying them. Models used to price mortgage backed securities during the financial crisis are a leading example.</p><p>	The use of “proxy data” for model input or output because the real data desired is unavailable, too expensive to obtain or cannot be objectively measured. Does sending out more e-mails with “creative phrases” mean that you are really a more creative and innovative person? FICO scores are a relatively good model for use in predicting credit risk, but not as a proxy when used for a whole host of other unrelated things such as predicting future job performance when used in the hiring process.</p><p>	Feedback loops, whereby a model increasingly encourages non optimal outcomes by rewarding certain behaviors at the cost of intended benefits. An example given is US college rankings which increasingly reward “user experience” and “research citations” rather than the actual educational outcomes for students.</p><p>	Algorithms that, mainly for efficiency purposes, use data from arbitrarily selected groups when individual data is available. Why should an algorithm price insurance for a driver based on the experience of other drivers that live near him or are in a similar economic position instead of on his own individual driving record?</p><p>	Optimization that is good for an algorithm’s owners but not for society as a whole. Monetary return is the most common priority for an algorithm used in the private sector, but is this what society wants when it results in micro-targeting the poor with the marketing of for-profit colleges that provide a below average education at great cost?</p><p>	Hidden biases and unfairness when assumptions are built into an algorithm that are reflective of social factors rather than individual experience. Algorithms used by the police to forecast crime are first used to predict easy to identify nuisance crimes which, unsurprisingly, occur mainly in deprived neighborhoods. The police have yet to develop an algorithm that forecasts where white collar crime takes place, although if they did Wall Street would surely light up red.</p><p>	Incorrect use of statistics; lack of feedback of model results into predicting outcomes. Baseball is a good field for prediction because outcomes – homes runs, strikes, batting averages - can be objectively measured and predictions can be fed back into the model to improve it. Using FICO scores in recruitment is not what the developers of FICO intended; there is no study suggesting statistical correlation between FICO and subsquent good job performance; good or bad job performance is also never used to assess HR models or improve their reliability.</p><p>The last chapter of the book looks at the micro targetting of voters with political messages through Facebook and similar social media sites. This was written in September 2016, a respectable amount of time before the world had a glimpse into the dizzying vortex of fake news, Russian hacking and tweeting Presidents. This is an excellent chapter and may well be seen as prophetic once we look back over our current period of political chaos, if it ever ends.</p><p>I resolved to write notes on the books I read as I get so much more out of them if I do. This resolution will last about one and a half books I’m sure, but if anyone is interested here they are:</p><p><b>Introduction:</b> Mathematical models are being increasingly used to make decisions that have real world impacts. However the algorithms they use are opaque except to a limited number of mathematicians or computer scientists and may, sometimes unknowingly, encode human prejudices, misunderstandings and biases.</p><p>Assumptions may be camouflaged by the maths and go untested and cannot be questioned by the persons to whom they are applied. These algorithms are often used in inappropriate contexts where there is insufficient objective data to properly apply the underlying statistical theory; algorithms may work for baseball with a many tens if not hundreds of thousands of objective, constantly updated data points but not to evaluate a teacher with a class of thirty. </p><p>The algorithms produce “feedback loops” under whereby they produce output that eliminates particular classes of result based on false assumptions, but as a result reinforce the weighting given the assumption in the model. Injustice is reinforced as these algorithms are applied to ever larger numbers of people, although the rich and privileged may still be assessed on an individual basis rather than as one of the masses assessed by machine. People unfairly denied opportunities as a result of this software are “collateral damage”.</p><p>The algorithms optimize a payoff designed by the developers of the model - in the case of many private sector models, monetary profit - but who is to say that this is the optimum payoff for society as a whole?</p><p><b>Chapter 1, Bomb Parts - What is a Model?</b> The author explains how her managing of her family cooking is a form of “data model”: inputs are family preferences, appetite that day; available food, special cases such as cooking on a birthday, the output is “family satisfaction” and the model determines menu for that day The model could be pre-programmed with a set of rules to determine its menu or could be trained through observing many examples. In either case mistakes may be made, perhaps through forgetting a rule or not including a rare case in the training data. The key point is that the model can incorporate personal biases that are not visible, in the case of the menu model towards healthy food and away from ice-cream.</p><p>Models don’t have to be complicated to be effective; a smoke alarm is a model intended to identify fire that operates on only a single input, the concentration of smoke particles. </p><p>Three questions are posited to evaluate models: Firstly, is the model understandable - or even visible to - by the people to whom it is applied? Secondly, does the model work in the subjects’ interests? Is it fair or can it cause unjust damage? Thirdly, does the model scale? May it be used in ever wider circumstances, at the same time reinforcing its hidden biases as it is applied in ever wider circumstances?</p><p>An example is the model used to determine sentence length in US courts based on a model about the risk of recidivism. The model takes into account factors, such as previous criminal record or age of first contact with the police, which would not be admissible as evidence in court and which may unfairly discriminate against certain sections of the population.</p><p><b>Chapter 2</b> talks about the role of algorithms in the financial crisis, noting two key issues that lead to the collapse in the mortgaged backed securities market. Firstly the assumption that models had been subject to proper mathematical vetting; in reality few people understood the mathematical and statistical issues and many that did lacked the integrity to speak up, especially as initial success had created its own feedback loop encouraging growth in the market. Secondly modern computing power had allowed a massive secondary infrastructure to grow around the market - credit default swaps, CDOs etc - that instead of diversifying the risk masked, magnified and concentrated it.</p><p>A very interesting point was made based on the author's experience in a risk assessment firm, that many traders are remunerated based on their Sharpe ratios, the ratio of revenues to risks taken, and accordingly are motivated to “...actively seek to underestimate…” risk in order to effectively manipulate the Sharpe ratio and hence their bonuses. This contrasts with the approach of hedge funds, which genuinely care about risks (given their own money is at risk) and traders at large financial institutions which don’t have their own financial capital at stake.</p><p><b>Chapter 3</b> looks at the impact of the algorithm developed by US News to rank colleges in the US. 75% of the ranking was based on proxy items intended to measure college success and 25% on subjective evaluation. The proxies selected, for example admission ratios, SAT scores, invited gaming that distorted applications; colleges would invest in sports in order to encourage applications that could be rejected; in an extreme case a new Saudi college required part time professors with large numbers of citations to change their site reference to the college in order to move up the rankings. The system does not measure the key success of education - what the students learnt at each school. The ranking generates its own feedback loop; colleges that rank high attract more applicants, generating more rejections thus moving them further up the rankings. Wealthy applicants pay consultants to game the system.</p><p>Crucially the original algorithm did not take into account college tuition costs. This guaranteed the early rankings being in line with existing “common sense” with Yale, Harvard and other wealthy colleges ranking high but thereby excluded an issue critical to students, value for money from an education, while encouraging colleges to spend excessively in order to improve student experience and hence ranking without regard to cost, ultimately leading to more student debt.</p><p>US govt has now made available data on schools allowing students to check directly.</p><p><b>Chapter 4</b> looks at targeted online advertising, taking for profit colleges that target poor and vulnerable people as a particularly nefarious case. The internet gives instant feedback for targeted marketing campaigns through Facebook or Google in particular where successful or failing ads can be evaluated in real time. Bayesian analysis is used to evaluate success.</p><p>20-30% of a for profit college’s budget may go on lead generation with more spent on recruitment than on education itself. Specialist lead generation firms exist targeting particular communities, posting fake job ads or promising health coverage.</p><p><b>Chapter 5</b> looks at “Predpol” an algorithm to predict crime which has a key input the geographic location of crime but which excludes data on race. The algorithm is successful in forecasting where crimes can occur, but mainly because it includes low level crimes - public drunkenness, jaywalking - which are better correlated with geographic location. However these crimes are also often associated with poverty and indirectly with race; more serious crimes are more difficult to detect while the system ignores some crimes altogether, such as white collar fraud. On the surface the algorithm is objective but under the hood it reflects value judgments around where the police direct their attention.</p><p>Issues of probable cause are also raised by the use of algorithms that predict whether a person will commit crime based on proxy data such as location of residence, whether or not employed or similar. These algorithms raise questions around to what extent the public is prepared to balance efficiency in police work against fairness, but without any public debate.</p><p><b>Chapter 6</b> looks at the use of algorithms in employee hiring decisions.. Personality tests are often used for screening job applicants, but these may be “run arounds” laws preventing discrimination against people with disabilities and, a key consideration for the correct use of algorithms, are rarely updated through monitoring their actual success in predicting good job performance. This may not be a problem for a system evaluating baseball stars, whose performance can be objectively statistically measured and where the individuals may be paid millions of dollars, rather the burden falls on lower paid staff in the collateral damage suffered by individuals that are screened out by the algorithms but are capable of doing the job.</p><p>These algorithms include negative feedback - those discriminated against fail to get good jobs, thus justifying the discrimination - and may be based on data that reflects historically discriminated hiring practices; an example is given of a system used in a British hospital that after many years of use was held to discriminate against women and immigrants, repeating discrimination reflected in the original data on which the system was based.</p><p><b>Chapter 7</b> examines the impact of algorithms in the workplace.</p><p>Job scheduling software uses data to schedule peaks and troughs of staffing, dehumanizing employees in the process who are unsure of their work schedule until only a day or two before being called on. </p><p>Another algorithm - ‘Cataphora’ - attempted to identify the most creative and innovative employees through tracking the flow of e-mail including certain key phrases through the e-mail system. There is little evidence that this approach worked, but employees who were not among those identified as the most creative may be first in line for termination. This algorithm suffers from the two classic problems identified early in the book, the difficulty of finding measurable proxy data for the items (soft skills such as creativity) that you want to measure and the lack of any feedback on success and failure of those measured to help the algorithm learn.</p><p>The chapter explained an egregious error in the evaluation of teacher performance through the use of SAT scores. The ‘Nation at Risk’ report issued by the Reagan administration was intended to address the decline in teaching standards as measured by a gradual decline in SAT scores of graduating students over the years. In fact the data illustrated an example of ‘Simpson’s Paradox’ in which the whole body of data showed one trend but when examined on a segmented trend the opposite trend was apparent. In this particular case the scope of people taking SAT tests had expanded over time including enough people at the lower end of the scoring range to lower the overall average (i.e. elite students had been sitting the test for many more years, so there was little room for scope to increase at the higher end of the scoring range). When the data was segmented in narrow SAT ranges SAT scores had increased in all segments over time. The whole premise of the report was false.</p><p><b>Chapter 8</b> looked at FICO scores. FICO scores themselves have some good features as data; they are based on an individual’s history (in contrast to being based on aggregate data inferred from people who resemble the individual), default on a loan is relatively objectively measured and feedback is used to improve scoring.</p><p>Problems arise when FICO scores are used as proxies for other data that is not so easily or objectively identified, for example future job success when used as screening in the hiring process, or when combined with other data in developing proprietary “e-numbers” used by firms for purposes such as marketing. These metrics are not transparent to consumers and risk being a backdoor for discrimination because they may use factors that are indirectly linked to poverty or race such as residence.</p><p><b>Chapter 9</b> considered how the whole insurance business model may be undermined by data algorithms. Insurance relies on the pooling of a wide range of risks, good and bad, to allow the pricing of the risk in aggregate. Algorithms allow such risks to be segmented in a non-transparent way which, in addition to undermining the insurance model itself by pricing out high risks, permits price gouging of other segments. The use of non risk related indicators, for example credit scores or location of residence, in pricing motor insurance also allows price gouging. Why should a drop in a driver’s credit rating impact his premium when the risk that he has an accident is unchanged? Drivers are being assessed on the consumer patterns of their friends and neighbors rather than on their accident record.</p><p><b>Chapter 10</b> looks at the application of algorithms to civic life, in particular the use of micro targeting of political messages through Facebook. Facebook has experimented with manipulating emotional responses of users through manipulating their newsfeeds. </p><p>Facebook’s approach to disseminating news is contrasted to a conventional newspaper in which the Editor makes a decision on what to put on the front page but that decision is seen by everyone and open to public debate whereas the criteria for selection of articles for a newsfeed by Facebook are opaque and unique to the viewer. Facebook micro targeting may be one of the reasons many Republican voters still believe Obama ‘birther’ and other conspiracies.</p><p>Micro targeting of voters risks disenfranchising everyone to the benefit of those paying for it. Voters in swing states are subject to more focused and intensive campaigns which are relevant only to themselves to the detriment of the democratic process for all.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/6816078?shelf=current-affairs"><span>current-affairs</span></a><span> </span><a href="https://www.goodreads.com/review/list/6816078?shelf=programming"><span>programming</span></a><span> </span><a href="https://www.goodreads.com/review/list/6816078?shelf=science"><span>science</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/35276649-amin"></a></section><section><span></span><span><p><span>396 reviews</span><span><span>405 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/2930182140">January 2, 2024</a></span></section><div><p><span>شاید بتوان عنوان کتاب را "سلاحهای کشتار جبری" ترجمه کرد تا منظور نویسنده بهتر حاصل شود، با این توضیح که منظور از جبر، معنای آن در ریاضی است و نه به معنای جبر و اختیار. فکر می کنم با محبوبیت این کتاب در دنیا و نقش رو به رشد داده پردازی در کشور ما دیر یا زود شاهد ترجمه آن باشیم.</span></p><p>بارها برای من پیش آمده که در فضای پیرامون که مملو از دانشجویان و محققین حوزه داده و مهندسی است بحثی پیش آید که مرزهای فنی و غیرفنی کار با داده چیست و تا چه حد پاسخ تمام مسائل به خودی خود از دل داده ها در می آید. یک بار کسی به من گفت دیگر نیازی به عامل انسانی نیست چون هوش مصنوعی خودش راه حل ها را می یابد و عرضه می کند. اما می دانیم که مثلا در کلان داده ها، به لحاظ آماری می شود بین پدیده های دلخواه بسیاری ارتباط معنادار یافت و جهت گیری محقق تعیین می کند که به چه ابعادی از آن معنا بدهد. به طور خلاصه اول مساله معنادار بودن نتایج را داریم و در مرحله بعد استفاده و تفسیر خاصی که از آنها می شود که در هر دو مرحله نقش عامل انسانی پررنگ است</p><p>این کتاب اما یک قدم هم فراتر می گذارد. الگوریتم هایی که نتایج تحلیل داده هایشان هم معنادار است و هم کاربردهای مشخصی دارد. اما گاهی خواسته و گاه ناخواسته معضلاتی درست می کنند که غالبا در راستای تخریب دموکراسی و افزایش فاصله طبقاتی هستند. با این که مثالها برای جامعه امریکا هستند، اما همین موقعیت جغرافیایی شاید بهترین مثال باشد تا وضعیتی رادیکال تر از استفاده از داده های بیشمار در تصمیم گیری های اجتماعی و سیاسی را ببینیم، البته با محوریت منافع اقتصادی و نقش شرکت ها. یک نکته مشترک هم در تمام این الگوریتم ها غیرشفاف بودن کارکرد آنها برای ناظران بیرونی و عدم وجود بازخورد برای اصلاح آنهاست که باعث شده نویسنده به هر کدام از آنها عنوان یک ابزار کشتار جبری بدهد</p><p>از سیستم رتبه بندی معلمان که گاه علیه معلمانی شایسته تر نتیجه می دهد و کل سیستم را بجای بهبود سیستم آموزشی به سوی افزایش امتیاز برای بقا در سییستم سوق میدهد، تا کارکرد شرکت های بیمه برای افزایش سودآوری که در نهایت به ضرر قشر آسیب پذیر عمل می کند، یا با همین منطق الگوریتم هایی که به پلیس کمک می کند تا خلافکاران بالقوه را شناسایی کند و در نتیجه زندان پر از افرادی می شود که غالبا از مناطق آسیب پذیر یا سیاه پوستان هستند. از این کاربردهای اجتماعی به سمت بازاریابی هدفمند که برویم پای دانشگاههای پولی و تبلیغات وسط می آید که به زبان نویسنده طبقه مرفه آگاه تر و باهوش تر از آن هستند که گول چنین سیستم هایی را بخورند؛ و در نهایت به سیستم اثرگذاری این الگوریتم ها در شبکه های اجتماعی میرسیم که اطلاعات دریافتی از سوی هر فرد را تحت تاثیر قرار می دهند تا جهت گیری سیاسی جمع زیادی از رای دهندگان مردد را قبل از زمان انتخابات تغییر دهند و به نوعی مانعی بر سر دموکراسی به شمار آیند</p><p>به نظرم فصل ها جذابیت های کم و زیاد متنوعی دارند. گاهی می توان با نظر نویسنده موافق نبود یا احساس کرد که در بیان نتیجه می تواند اغراق کند، اما در کل این کتاب را اثری خواندنی یافتم که با تخصص خود نویسنده که در همین باره است قابل اعتنا می شود</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/35276649?shelf=business"><span>business</span></a><span> </span><a href="https://www.goodreads.com/review/list/35276649?shelf=politics"><span>politics</span></a><span> </span><a href="https://www.goodreads.com/review/list/35276649?shelf=science"><span>science</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/2199404-andy"></a></section><section><span></span><span><p><span>1,682 reviews</span><span><span>547 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/2151100186">May 7, 2022</a></span></section><div><p><span>It's good to critique what the author calls WMDs, but for me this book missed the mark. </span></p><p>For example, the entire example of teacher-rating in DC is a red herring. The real question is: What is proven to deliver success in an urban US school system? (Answer: <a href="https://goodreads.com/book/show/6402713.Hope_and_Despair_in_the_American_City_Why_There_Are_No_Bad_Schools_in_Raleigh">Hope and Despair in the American City: Why There Are No Bad Schools in Raleigh</a>) So it's valid to point out the silliness of a silly GIGO teacher-rating algorithm, but it's a distraction to fall down the vortex of dissecting that silliness in detail. We need to focus instead on facts about relevant outcomes. This is sorta kinda vaguely the theme the author seems to be developing but never fully formulates: the afterword delivers a pretty wishy-washy conclusion about the nature of objective truth. </p><p>Trying to say something nice... The flip side of the superficiality of this book is that it alights on numerous interesting factoids as it skims along. Also, it is clearly written.</p><p>Alternatives: <br/><a href="https://goodreads.com/book/show/1301.Moneyball_The_Art_of_Winning_an_Unfair_Game">Moneyball: The Art of Winning an Unfair Game</a><br/><a href="https://goodreads.com/book/show/26889576.The_Big_Short_Inside_the_Doomsday_Machine">The Big Short: Inside the Doomsday Machine</a><br/><a href="https://goodreads.com/book/show/15811520.Detroit_An_American_Autopsy">Detroit: An American Autopsy</a><br/><a href="https://goodreads.com/book/show/1301.Moneyball_The_Art_of_Winning_an_Unfair_Game">
  
</a><br/><a href="https://goodreads.com/book/show/26889576.The_Big_Short_Inside_the_Doomsday_Machine">
  
</a><br/><a href="https://goodreads.com/book/show/15811520.Detroit_An_American_Autopsy">
  
</a></p><p></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/3079724-chad-kohalyk"></a></section><section><span></span><span><p><span>295 reviews</span><span><span>31 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1767214145">October 3, 2016</a></span></section><div><p><span>Solid overview of the various mathematical models that govern our education, labour, wealth, and commerce. O'Neil packs in many examples and unpacks how simplistic, unfair and damaging to already disadvantaged people these models can be. As someone who worked on the front lines of developing models for predictive internet shopping, I was familiar with many of the tactics mentioned in this book, and their ethical shortcomings (which finally led to me leaving the business). What she says is entirely true, and it makes a small amount of people a lot of money. More people <em>should</em> be outraged. This book could help.</span></p><p>There are two shortcomings, though, both possibly due to its short length. First, there is no discussion at all of government abuses of all the data collection detailed within its pages. Maybe in the age of Snowden this is just assumed, but I think it is an important outcome of the big data revolution that should at least be addressed, even in passing. Secondly, I was hoping for more imaginative solutions. The <a href="https://en.wikipedia.org/wiki/Financial_Modelers%27_Manifesto">"hippocratic oath" for data scientists she refers to</a> is a nice idea... from twenty years and two big economic recessions ago. Advocating for stronger regulation is certainly prudent. However, I was hoping for something new.</p><p><em>Weapons of Math Destruction</em> is a well documented tour of the standard examples of the misuse of math and big data, and concludes with the standard solutions. A good book to recommend to friends who need a primer on these issues that we have been facing for the last decade or so.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/3079724?shelf=audio"><span>audio</span></a><span> </span><a href="https://www.goodreads.com/review/list/3079724?shelf=general-nonfiction"><span>general-nonfiction</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/5648532-atila-iamarino"></a></section><section><span></span><span><p><span>411 reviews</span><span><span>4,445 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1890702715">February 7, 2017</a></span></section><div><p><span>Uma discussão muito boa e ponderada sobre o que acontece quando começamos a usar algoritmos para avaliar pessoas. A <a href="https://goodreads.com/author/show/6928121.Cathy_O_Neil">Cathy O'Neil</a> passa por vários sistemas de avaliação automatizados e o tipo de viés que eles introduzem, quem fica de fora ou quem é mau avaliado. De sistemas de crédito a seguros, saúde, vagas de emprego e antecedentes criminais. Uma visão bastante importante do que está por vir com o dataísmo crescente (ver <a href="https://goodreads.com/book/show/31138556.Homo_Deus_A_Brief_History_of_Tomorrow">Homo Deus: A Brief History of Tomorrow</a>).</span></p><p>Não que isso seja um problema novo. Preconceito, premissas erradas e mais uma série de falhas desfavorecem muita gente desde sempre. Mas agora que podemos automatizar essa avaliação, o que eram vários humanos errando em várias direções vira um algoritmo errando consistentemente da mesma forma. O que pode aumentar o estrago feito, mas também dá margem para um aprimoramento muito mais eficiente.</p><p>A mensagem mais importante do livro me parece ser que, já que o uso dos nossos dados para tudo é inevitável, pelo menos saber o que está sendo levado em conta quando somos avaliados ajuda muito a previnir erros com consequências graves.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/5648532?shelf=dataismo"><span>dataismo</span></a><span> </span><a href="https://www.goodreads.com/review/list/5648532?shelf=futuro"><span>futuro</span></a><span> </span><a href="https://www.goodreads.com/review/list/5648532?shelf=tecnologia"><span>tecnologia</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/41236164-jack-teng"></a></section><section><span></span><span><p><span><span><span>Author <i></i></span></span></span><span><span>8 books</span></span><span><span>7 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1776309459">October 7, 2016</a></span></section><div><p><span>I can't stress how important of a book this is. I don't think people really know how the obsession with Big Data and algorithms is about to control/influence our lives. I suppose I sound paranoid, but I really don't think I am. There are too many tinkerers out there who have some degree of competence at math and who think they'll solve all the world's problems with the next greatest optimization formula, and yet they lack even the most basic experience in asking proper research questions and understanding the major limitations of data. My own experience in theoretical ecology exposed me to empiricists wanting to get in on the modeling action and just produced garbage papers. Beware, people. Beware.</span></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/9656259-c-p-cabaniss"></a></section><section><span></span><span><p><span><span><span>Author <i></i></span></span></span><span><span>7 books</span></span><span><span>112 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1703013978">February 20, 2017</a></span></section><div><p><span>*I received a copy of this book through Netgalley. All thoughts are my own.*</span></p><p>This book did not turn out to be what I was expecting. I expected O'Neil to go more in depth about the math behind data she was discussing, to explore the algorithms in greater detail. That was not what I got, however. This book turned out to be more of a superficial look at some of the ways big data can/has impacted society in the author's opinion. And while I found some of the material presented interesting and informative, I was overall not impressed by the arguments the author made. She didn't seem to have a clear idea for a resolution. </p><p>More thoughts will be up on my blog: <a href="http://www.courtneysreads.blogspot.com">www.courtneysreads.blogspot.com</a></p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/9656259?shelf=nonfiction"><span>nonfiction</span></a><span> </span><a href="https://www.goodreads.com/review/list/9656259?shelf=reviewed"><span>reviewed</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/13716226-murtaza"></a></section><section><span></span><span><p><span>686 reviews</span><span><span>3,390 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/2247876576">January 21, 2018</a></span></section><div><p><span>A short and concise overview of the problems being wrought by the algorithms that are now quietly governing our society. While the overview is useful, I was a bit surprised at how little new information that there was in here. The issues that she raises should be familiar to anyone who has been following the impact of Big Data in even a cursory way. Facebook news bubbles, insurance profiles and other common phenomena are the main examples she uses. I did appreciate the social justice focus of the book, which makes sense given that the author is a former hedge fund analyst disenchanted by the malign behavior that she witnessed on Wall Street. </span></p><p>One key takeaway is that it there is a drive for poor and middle class people to have their lives processed by algorithms, while the rich will continue to depend on real-life personal connections to get by. In effect the masses are being turned into raw numbers with scores attached, while a small elite will continue to be authentic "personalities" who live in the more nuanced way. </p><p>This book lays out some of the ways to fight back against opaque algorithmic power and make these algorithms more fair. A solid overview but nothing groundbreaking.</p><p></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/5861086-margaret-sankey"></a></section><section><span></span><span><p><span><span><span>Author <i></i></span></span></span><span><span>9 books</span></span><span><span>233 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1759513938">September 16, 2016</a></span></section><div><p><span>I was really delighted to see this book on the list of nominations for the National Book Award--O'Neil was a math professor, then hedge fund analyst, then Occupy activist, now excellent non-fiction explicator of the power that we've given algorithms in daily life. Certainly, the old way, in which a human bank manager or college admissions officer selected people for loans or Harvard reflected significant prejudices--but have we gone too far in the other direction, thinking that hard data will erase biases? O'Neil offers vivid examples: microtargeted ads, teacher evaluations, police stats for predictive patrols, insurance, credit scores and the warping of higher ed around the US News and World Reports college rankings to demonstrate the potential for toxic grips of big data as opaque, unexplained, unchallenged and dictatorial, as well as the possibilities for using its power to help rather than exclude.</span></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/3502690-graeme-roberts"></a></section><section><span></span><span><p><span>522 reviews</span><span><span>36 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1980462570">November 3, 2021</a></span></section><div><p><span></span></p><p><a href="https://goodreads.com/author/show/6928121.Cathy_O_Neil"> Cathy O'Neil</a> writes well and provides some good information, but I found the fundamental thesis of <a href="https://goodreads.com/book/show/28186015.Weapons_of_Math_Destruction_How_Big_Data_Increases_Inequality_and_Threatens_Democracy"> Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy</a> to be foolish and irresponsible. The title, condensed inevitably to WMD, is used promiscuously to describe any application of data science, statistics, or even information technology that she considers to be unfair or discriminatory. In some cases, she may be right; some applications are used by the greedy and the uncaring with insufficient thought of the consequences, but she rarely considers the positive attributes of the applications. Once branded as a WMD, a term indelibly appended to the evil trifecta of George W. Bush, Saddam Hussein, and the Iraq War, how are readers meant to consider their many positive aspects and what can be done to make them better?</p><p>At first, I thought that Ms. O'Neil had been the victim of a foolish PR person in her publishing company who had come up with this catchy name, but her ardent use of the term convinced me that she had invented it.</p><p>Ms. O'Neil's own career seemed to follow a downward trajectory from her Harvard PhD in mathematics and a teaching position at Barnard to a leading hedge fund and a series of other ethically questionable environments that left her convinced that data science, and even mathematics, are roots of much evil. I don't buy it. They are tools that can be used or misused, and her diatribe does nothing to help.</p>She asserts, on page 210:<blockquote>
  <p>If we find (as studies have already shown) that the recidivism models codify prejudice and penalize the poor, then it's time to have a look at the inputs. In this case, they include loads of birds-of-a-feather connections. They predict an individual's behavior on the basis of the people he knows, his job, and his credit rating—details that would be inadmissible in court. The fairness fix is to throw out that data.</p>
  <p>But wait, many would say. Are we going to sacrifice the accuracy of the model for fairness? Do we have to dumb down our algorithms?</p>
  <p>In some cases, yes. If we're going to be equal before the law, or be treated equally as voters, we cannot stand for systems that drop us into different castes and treat us differently.</p>
</blockquote>I disagree. Some data is better than none (provided it is accurate and not intentionally manipulative), though we should be looking to constantly improve our data science systems. I would like to know that the recently paroled criminal moving in next door had been released by a judge in full possession of whatever facts are available. So, I suspect, would Ms. O'Neil.<p></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/8681481-lauma"></a></section><section><span></span><span><p><span>224 reviews</span><span><span>1 follower</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1788643896">January 29, 2021</a></span></section><div><p><span>Datu zinātnieka obligātā literatūra.</span></p><p>Interesants ieskats dažādos matemātisko modeļu piemēros, kas cilvēku pieļautu kļūdu vai alkatības dēļ pasaulei dod vairāk slikta, nekā laba. Jebkurš modelis ir tik subjektīvs, cik subjektīvs ir tā veidotājs. </p><p>Mašīnām var iemācīt diskrimināciju.<br/>Cilvēki jauc korelāciju ar cēloņsakarību.<br/>Cilvēkiem patīk pelnīt uz citu cilvēku rēķina.</p><p>Autore norāda, ka bīstami matemātiksie modeļi ir mums apkārt un kontrolē mūsu dzīves daudz vairāk nekā mēs iedomājamies. Augstās mācību maksas ASV universitātēs? Pie vainas izglītības iestāžu tops, kurā samaksa par studijām nav iekļauta. Matemātiskie modeļi, kas paredz noziedzību pirms tā vēl notikusi? Ideja lieliska, izpildījums pieklibo, jo policistiem ir tieksme fokusēties uz maziem pārkāpumiem, kas bieži norisinās nabadzīgajos rajonos, kurus galvenokārt pārstāv ne tās gaišākās ādas krāsas pārstāvji. Jo vairāk tādiem pievērš uzmanību, jo vairāk modelis domā, ka tur notiek visaktīvākā noziedzība, kamēr bagāto un 'pieklājīgo' cilvēku rajonā vardarbību ģimenē gadiem neviens nepiefiksēs. Demokrātija? Cilvēkiem pielāgotas reklāmas for the win - dažādi cilvēki saņem dažādus solījumus no viena un tā paša politiķa. Facebook arī spēj ietekmēt cilvēku emocijas pietiekami, lai tas varētu izšķirt vēlēšanu dienu rezultātu. Vēlēšanu rezultāti nav jāvilto, pietiek ar pareizu cilvēku manipulāciju e-vidē, un matemātiskie modeļi to spēj nodrošināt.</p><p>Piemēru ir daudz, taču visus sliktos modeļus vieno 3 svarīgas īpašības - tie nav caurredzami, tie ietekmē lielas cilvēku grupas un to ietekme ir spēcīga ne tajā labākajā nozīmē. Nav tā, ka paši matemātiskie modeļi ir slikti, tie ir tikai instruments cilvēku rokās, un jebkurš instruments ir tik labs, cik labs un spējīgs ir tā izmantotājs. Veltot pietiekami daudz uzmanības sava darbam, jebkurš datu zinātnieks var parūpēties par to, lai viņa veidotais modelis nenonāktu šo piemēru sarakstgā un neradītu kaitējumu cilvēcei. Arī man ir gadījies upurēt modeļa precizitāti, no tā izņemot cilvēku radīto diskrimināciju datos. Jāatzīst, ka to izdarīt nemaz nav tik viegli, ja esi mācījies tiekties pēc iespējami precīzākā iznākuma, taču ir jāsaprot, ka ne vienmēr augstākais % skaitļos atbilst realitātei.</p><p><i>Sometimes the job of a data scientist is to know when you don't know enough.</i></p><p>Iesaku arī tiem, kas īpaši nedraudzējas ar matemātiku. Grāmata uzrakstīta pietiekami saprotamā valodā un vairāk iedziļinās tajā, kā strādā pasaule, kurā dzīvojam, pārāk neiedziļinoties matemātisko modeļu tehniskajās detaļās.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/8681481?shelf=math-data-science"><span>math-data-science</span></a><span> </span><a href="https://www.goodreads.com/review/list/8681481?shelf=nonfiction"><span>nonfiction</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/2704424-nick-klagge"></a></section><section><span></span><span><p><span>779 reviews</span><span><span>65 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1832817871">February 5, 2017</a></span></section><div><p><span>Cathy O'Neil was my professor for number theory in college, I think in 2006. I thought she was a great teacher, but didn't keep in touch at all after the class. I was somewhat aware that she was involved with Occupy Wall Street's financial policy arm, and after I heard about this book, also learned that she had been co-hosting a podcast on Slate (which she is now about to leave!--but I still have a lot of back-episodes to listen to).</span></p><p>I'm broadly in agreement with the thrust of her argument in this book, and only have quibbles with regard to ways I think it necessarily simplifies things to be a popular-consumption nonfiction book. The main thrust of the book is that people frequently conflate "algorithmic decision-making" with "neutral decision-making", and that this is a fallacy (one that algorithm-purveyors are happy to perpetuate, for the most part). As "big data" and quantitative models get applied to more and more aspects of everyday life, it's incumbent on us to understand this, and to consider ways in which algorithmic decision-making can be problematic, biased, or dangerous. </p><p>O'Neil describes three features that characterize a "weapon of math destruction": scale, secrecy, and destructiveness. We mostly only need to focus on algorithms that have all three of these features, at least to some degree--a model that is in limited use, transparent, or harmless is not much of a cause for concern. She gives examples of algorithms in many fields that she sees as meeting all three of these criteria. One example is recidivism risk modeling, which is now used in many states to determine, at least in part, criminal sentencing. Errors or bias in these algorithms may result in additional years behind bars for the individuals they apply to, and they are both widespread and not publicly disclosed. There are many other interesting (/troubling) examples in the book, such as teacher value-added models.</p><p>An emergent property of many such algorithms is that they may engender undesirable feedback loops. For example, a recidivism risk model will be biased against black people if it is trained on historical data that cover an environment characterized by bias against black people. (If black people are more heavily prosecuted in general, they are likely to appear as higher recidivism risks, and even if the algorithm doesn't use race directly, it will pick up on correlated factors and amount to the same thing.) This is largely a function of the opacity characteristic--if an algorithm is publicly disclosed, people can bring scrutiny to it and highlight flaws.</p><p>The issue of model opacity is an especially interesting one to me, as someone who works on regulatory financial models that are intentionally not disclosed. There are strong reasons for not disclosing any models used for high-stakes decisions (in my case, setting minimum capital levels for banks). A primary concern is that a transparent algorithm will be "gamed", in the sense that those subject to it will figure out ways to make themselves "look good" to the algorithm that are driven more by the details and limitations of the model rather than by the underlying substance. A second concern is that a transparent regulatory model can encourage a "monoculture" in which those subject to it will simply adopt the model for themselves, rather than developing their own models that, while still flawed, will have different flaws than the regulatory model.</p><p>I don't think there is an obvious solution to this transparency problem. One solution that I definitely don't agree with is to eschew quantitative decision-making altogether. As O'Neil clearly states in the book, we shouldn't assume that pre-algorithmic decision-making was unbiased either--it seems quite apparent that, for example, there is bias in judgmental sentencing, perhaps more than in algorithmic sentencing. O'Neil herself has one proposed solution to this (which I don't think she really discusses in the book)--she has started a consulting company whose intent is to audit existing algorithms for potential biases or other damaging impacts. This would allow some degree of independent assessment while not disclosing the model generally. I think this is an interesting idea and I hope it takes off, but there are limitations. Especially in the private sector, it's not clear why a company would voluntarily request such an audit, especially if an algorithm is making them lots of money. Fear of regulatory penalties could be one motivation, but we're clearly entering an era of deregulation. Regulators themselves might force audits, but again that requires a strong regulator (also, who watches the watchers?). </p><p>One approach that I think could make sense in at least some contexts is a hybrid algorithmic-judgmental process. (At least right now, hybrid processes can be most effective in many fields--for example, the best chess player is neither a computer nor a human, but a human assisted by a computer.) To take the example of recidivism risk, we might have an algorithm that is publicly disclosed that produces a publicly-disclosed outcome to the judge (perhaps a recommended range). The judge may then choose to depart from the recommendation, but needs to give a written description of her reasoning for doing so. In this way, the algorithm can be audited by any outside party for potential biases, but the final judgmental step serves as insurance against flagrant cases of gaming the system, or cases with significant factors that are not considered by the algorithm.</p><p>As I said earlier, my only real quibbles with the book are around simplifications that I think are a reality of publishing a non-fiction book for popular consumption. For example, the terminology of "WMD" encourages us to think in a binary way (is it or isn't it one?), rather than seeing a continuum from OK to troubling, which I think is a better reflection of reality.</p><p>Finally, I'll add that this book, published in September 2016, proved to be quite prescient. The public controversy about "fake news" and the Facebook newsfeed algorithm arose shortly after its publication. Interestingly, I think before this happened, few would have identified the newsfeed algorithm as a potential WMD, because the vector for "destructiveness" was non-obvious. O'Neil has written some articles on this topic since the publication of the book, which are worth looking up.</p><p></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/1444762-jill"></a></section><section><span></span><span><p><span>2,203 reviews</span><span><span>93 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1748029813">September 4, 2016</a></span></section><div><p><span>This book shows the hidden ways in which the use of "Big Data" is much more far-reaching and harmful than expected. Big data refers to the massive amount of information now available because of computers that is collected and analyzed and sold to third parties.</span></p><p>In particular, as the author demonstrates convincingly, applications of Big Data “punish the poor and the oppressed in our society, while making the rich richer.” She paints a sobering picture.</p><p>The author calls the mathematical models employing Big Data and used to such harmful effect “Weapons of Math Destruction” or WMDs. </p><p>In WMDs, she explains, “poisonous assumptions . . . camouflaged by math go largely untested and unquestioned.” They create their own toxic feedback loops, and, to an extent which shocked me, guide decisions in a large variety of areas ranging from advertising to prisons to healthcare to hiring and firing decisions. Most importantly, because they rely on esoteric mathematical models, no matter that they are many times based on biased and/or erroneous premises:</p><p>“They’re opaque, unquestioned, and unaccountable, and they operate at a scale to sort, target, or ‘optimize’ millions of people.”</p><p>The goal is always profit, but what is lost is fairness, the recognition of individual exceptions, and simple compassion and humanity, adding to the inequality gap, not to mention downward spirals for some unfortunate victims from which it is almost impossible to escape.</p><p>I am not at all well-versed in math, but the author manages to explain how all this works without requiring that one understand specific algorithms. She provides examples from the worlds of teacher evaluations, hiring decisions generally, advertising, insurance, police programs, college admissions, lending and credit evaluation, and political targeting. </p><p>One of the saddest chapters (and they are all sad, unfortunately) is about the many for-profit universities (Trump University comes to mind) that specifically target people in great need, selling them overpriced promises of success. Her quotes from the marketing materials of these places are horrifying. They look for individuals who are “isolated,” with “low self esteem” who have “few people in their lives who care about them” and who feel “stuck.” She shows how they use google searches, residential data, and Facebook posts, <em>inter alia, </em> to find “the most desperate among us at enormous scale”:</p><p>“In education, they promise what’s usually a false road to prosperity, while also calculating how to maximize the dollars they draw from each prospect. Their operations cause immense and nefarious feedback loops and leave their customers buried under mountains of debt.”</p><p>The chapter on the way the “stop and frisk” policing operates is also very depressing; and in truth we have seen the tragic results in city after city.</p><p>The fact is, the whole book is rather a downer, albeit an important one. Although O’Neil cites a few programs that have used Big Data to help people rather than to enrich a few and oppress the rest, can one really think that “moral imagination” can take precedence over prejudice and greed? Personally, I’m not so sure. The author provides ideas about how to change (and importantly, regulate) uses of Big Data, but she is more optimistic than I am, ending on a positive note:</p><p>“We must come together to police these WMDs, to tame and disarm them. My hope is that they’ll be remembered, like the deadly coal mines of a century ago, as relics of the early days of this new revolution, before we learned how to bring fairness and accountability to the age of data. Math deserves much better than WMDs, and democracy does too.”</p><p><strong>Evaluation:</strong> I hope this important book gets a lot of attention. My husband always makes the argument about privacy concerns that what do <em>we</em> care if we’ve done nothing wrong? This book shows how, astoundingly, that isn’t enough to stop Big Data from hurting us in many aspects of our lives. It is a critical lesson for today’s world, and the world of our children.</p><p></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/3938672-kurt-pankau"></a></section><section><span></span><span><p><span><span><span>Author <i></i></span></span></span><span><span>11 books</span></span><span><span>21 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1751883938">November 1, 2016</a></span></section><div><p><span>My father, in one of his grouchier get-off-my-lawn moments, complained to me about computers in the workplace. Specifically, it bothered him that people were so trusting in the software that they were unwilling to gut-check outputs. This conversation happened maybe twenty years ago, but it stuck with me, and part of the reason is because he was absolutely right. We've all heard the phrase "garbage in, garbage out" to describe this phenomenon.</span></p><p>O'Neil takes the idea of "garbage in, garbage out" and compounds it, and this book is an exploration of the pitfalls, both real and potential, involved in letting algorithms dictate our lives, in everything from teacher retention to prison recidivism to workplace wellness programs. I'm a huge fan of O'Neil's contributions to the Slate Money podcast, which is how I learned of this book in the first place. And if you're at all interested in the Big Data economy, you should go read it right now. Even if you're afraid of math--in fact, especially if you're afraid of math. O'Neil's writing is not overly technical. This book is a very easy read and it is dense with ideas.</p><p>Many of these ideas resonated the instant they hit my brain, making me recognize things that had been around me for months or years. To give an example, O'Neil has a chapter on insurance and describes how more insurers are using e-scores based on "birds-of-a-feather" logic and demographic information. She points out the extent to which these things can become ridiculous, where having a drunk driving conviction can result in LOWER car insurance premiums than having a bad credit rating. While reading this, I remembered the "Sorta Marge" commercial Esurance put out and thought "Oh, that's what that was about."</p><p>There's a big social justice component to this book that hinges on the fact that algorithms sacrifice individuals in favor of trends, and in doing so they create feedback effects. Another example: some states include the defendant's zip code in sentencing models. This would be inadmissible in a court case, because it's a proxy for race and/or income and is irrelevant to the individuals guilt or innocence. But sentencing modelers argue that it accurately predicts how likely they are to be a repeat offender. So a person from the "wrong" zip code gets a longer sentence, then it's harder for him/her to find a place in society after release, and they end up going back to jail. The model doesn't just predict, it reinforces, and this is a problem. Because inherent in these models is an idea that O'Neil circles back on again and again: since we use these models to help shape the future, they have to reflect the kind of future we want to live in rather than just codifying the past.</p><p>It's another book I desperately want everyone I know to go out and read so we can have lengthy discussions about it in bars. Here's my only real gripe: I hate the phrase "Weapons of Math Destruction." It's an excellent title and a key concept in the book, but O'Neil uses "WMD" as shorthand throughout. Every time I saw it, I had to remind myself that it meant "Math" destruction instead of "Mass" destruction, and by about Chapter 5 the joke had stopped being funny to me. End rant.</p><p>I loved this book and I will no doubt be reading it again. I could talk about it for a long time, so I'm just going to stop. But, as O'Neil points out, we're in the early days of Big Data. These mathematical models are affecting your life whether you realize it or not, and they don't have to be pernicious. But now is the time to start informing yourself if you want to help shape the conversation tomorrow.</p><p></p></div><section><ul><span></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/53069903-wick-welker"></a></section><section><span></span><span><p><span><span><span>Author <i></i></span></span></span><span><span>7 books</span></span><span><span>520 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/3550541318">December 9, 2020</a></span></section><div><p><span><b>Big data codifies prejudice and penalizes the poor.</b></span></p><p>A decent take on the "garbage in-garbage out" theory of data utilization. Cathy O'Neil appears aptly qualified to comment on the phenomenon of <b>data models becoming an amplified version of societies biggest problems.</b> O'Neil has invented her own term, <b>Weapons of Math Destruction (WMDs) to lift the veil on a hidden but very active phenomenon that is perpetrating the racial and class biases that are already prevalent.</b></p><p>Per O'Neil, WMDs are any data algorithm that is used by any governmental or private organization to increase output, decrease waste and offload the critical thinking from humans to computers. <b>The problem with WMDs is that they are not dynamic models that adapt and change based on a feedback loop. </b>Instead, old data is feed into these machines which then crunch numbers and output <b>a result that is considered sacrosanct by whoever is using the algorithm.</b> The WMD almost becomes like a mystic--you cannot question the WMD, it is beyond repudiation. Yet the output is opaque and the result compounds existing problems.</p><p>Examples abound. School districts use WMDs to determine who the "bad teachers" are to then get rid of them. Yet these programs will score one teacher at the bottom one year and at the top the next year in an arbitrary way. The issue is that a teacher has only a few number of students, not say 10,000. <b>These low numbers make the data amplify noise, not signal.</b> Other models like baseball statistics can be effective because it is wholly transparent, where WMDs are not, using proprietary algorithms. <b>Bad models are opaque, blind to pertinent information and are not dynamic.</b></p><p>Data models are a black box beyond scrutiny and have a positive feedback loop within themselves that serves as self prophecy. The US world and News report offers a great example of becoming the standard upon which college data models have been fit. <b>The metrics become the target</b>, completely invalidating the list to begin with. The cost of education isn't put into the formula which creates a "gilded checkbook" for colleges. <b>Tuition has sky-rocketed as a result of this phenomenon. Predatory for-profit colleges take advantage of the vulnerable by using WMDs on social media.</b></p><p>Geography used in crime WMDs are a surrogate for poverty and race because cities are highly racially segregated to begin with. <b>Data models for lending, insurance and employment based on zip code amplify racial wealth disparities. Prison recidivism rates based on WMDs have the exact same problem.</b> Scheduling practices for workers using WMDs create unstable work life balances for the impoverished. <b>Credit scores used as proxies for hiring compound the positive feedback loop of poverty and debt.</b> Credit is a marker of wealth and wealth is a marker of race. All of these data go into WMDs and amplify the problems once again. There is asymmetry in political micro targeting thanks to WMDs.</p><p>Overall, informative, if not a tad outdated (written 2016) read about the misuse of bad data. I recommend.</p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/53069903?shelf=nonfiction"><span>nonfiction</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/5935491-tanja-berg"></a></section><section><span></span><span><p><span>2,051 reviews</span><span><span>491 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/2306703760">February 25, 2018</a></span></section><div><p><span>Much of our lives are now influenced by algorithms. These are deemed to be neutral, but in fact many are based on models that are glaringly discriminatory. Zip codes stand in for race, for example. The poor are being particularly targeted for pay-day loans. Teacher are measured on opaque proxies that do no reflect their skills at all. Your credit score is as much a result of the group of people you are thrown in with, as with your own behavior. These algorithms are what the author calls "weapons of math destruction". </span></p><p>Every on-line decision you take, every click you make, is being monitoried. Your ads are tailored, and so is your Facebook news feed. "Big brother sees you" has a whole new meaning since "1984" was written 50 years ago. We willingly leave a trail of information across a range of sites. We take stupid tests on FB that sells information on ourselves and our friends to the highest bidder. Some times it seems that you only need to consider making an investment for the ads to start trailing your feed and your online existence.</p><p>Turns out that in making hiring decisions, intelligence tests are prohibited - that is why personality tests flourish. Personality is being used as a proxy for intelligence. This despite the fact that there is little correlation between personality and job performance and that only one trait - conscientiousness - bears any correlation to work performance at all. Job seekers aren't told when their personality test red-flags them even for minimum wage jobs. </p><p>This book reinforces my notion that we're all screwed. Still, awareness that the internet bubble we all live in does not represent the ultimate truth, would be beneficial. </p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/5935491?shelf=non-fiction"><span>non-fiction</span></a><span> </span><a href="https://www.goodreads.com/review/list/5935491?shelf=science"><span>science</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/7494007-aaron"></a></section><section><span></span><span><p><span>203 reviews</span><span><span>43 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1739469120">March 3, 2017</a></span></section><div><p><span>We've all seen the Big Data books: the future is now! A/B testing forever! AlphaGo crushed it! OkCupid says you shouldn't have a shirtless fish pic, you adorably dull redneck!</span></p><p>But Big Data has a darkside, and O'Neil goes through each segment of our life to show how these "models" can be used against us, to extract goods from us, and to keep us poor. Unfortunately, she also loses her argumentative power that could come with nuance, and she has to disregard nuance in order to make it understandable to the layperson (in other words, I don't think she's very charitable to the layperson). </p><p>The first big issue she brings up and has lots of evidence for is: Transparency. Lots of Big Data models that we use (and are fed into) on a daily basis, are not transparent. They're opaque equations sitting on a server farm. A teacher being graded on a value added model doesn't know where their score is coming from. A potential hire doesn't know why he failed his psyche evaluation. A criminal standing before judge doesn't know why the score said he'd be more like to return to jail. </p><p>This is honestly one of my biggest takeaways, and it hits close to home. When you apply for a credit card, they tell you why you failed to get one. You can access your "data point" and know why you have the FICO score you do. Google and Facebook tell you the metrics they use to serve you ads in their settings. I know that Amazon is trying to get me to buy another wallet even though I just bought a wallet because they showed me five hundred ads for another wallet. But what about those ads that Forbes tries to serve me? The cookies sitting on my computer watching me? I have no ideas what they're doing and not readily known way to find out. What about non-regulated credit systems that exist out there in Web 2.0 land? Bank of America controls for race and tries to stop redlining when they make a new policy, but will Peter Thiel try to do that when he invests in E-Corp? Probably not.</p><p>The second big problem with some Big Data systems is that they create feedback loops that increase inequality. Here, O'Neil is super weak except when she brings up the criminal justice example- we could be using big data to help keep people out of prison and make programs that lower recidivism, but instead we're using it as a way to keep white people out of prison.... but am I really supposed to believe that ads help make poor people poorer? She brings up for-profit schools using targeted ads to lure immigrants and poor people into massive student debt to make a profit, and, while I admit that's super shady, it's not the targeted ads' fault, is it? </p><p>The third problem with these "Weapons of Math destruction" is that they often have skewed data. This is the old line "garbage in, garbage out" except now it's "racist/sexist garbage in, racist/sexist garbage out." For example, if you make a employment system that filters out resumes, than teach it on a bunch of older resumes, you're inputting the bias of those older resumes. So if the guy that was reading those resumes was racist, you might be teaching a racist model.</p><p>Technically O'Neil has two other "bad points" about "WMDs" but they're just about scale.</p><p>Now, there are a lot of problems in this book and O'Neil kind of goes on tangents. For one thing, the WMDs she brings up are less "weapons" than they are symptoms of a bigger societal problem. Take "democracy": our current political system allows a few people- those that live in Orlando, Florida and Pennsylvania, basically- to chose who will be the President. This is messed up, but it means that the Democratic Party could build a powerful machine learning system that most efficiently spent money in locations to help change hearts and minds and win. She really dislikes this Big Data system, and says it's a threat to democracy...</p><p>... but the electoral system itself is giant problem and threat to democracy (see: election of 2000)! Big Data has nothing to do with it!</p><p>The book ought to have been longer, and it ought to have included more counterexamples of positive data models (I can recall only two, FICO and some housing model). I think that she should've, if not had hand written equations or step-by-step instructions, at least given some background on actual data science. The way it is written makes it seem like she's a magician-mathematician that wandered down from the Ivory tower and realized that bankers were using magic for evil and now she wants to raise hell. </p><p>But I guess if I wanted authors to stop writing popular non-fiction books that they A/B tested on their blogs and turned into TED talks, I should stop reading popular non-fiction. </p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/7494007?shelf=non-fiction"><span>non-fiction</span></a><span> </span><a href="https://www.goodreads.com/review/list/7494007?shelf=statistics"><span>statistics</span></a></span><span></span></ul></section><hr/></section></article></div><div><article><div><section><a href="https://www.goodreads.com/user/show/1459884-krysten"></a></section><section><span></span><span><p><span>492 reviews</span><span><span>20 followers</span></span></p></span></section></div><section><section><span><a href="https://www.goodreads.com/review/show/1760614208">September 29, 2016</a></span></section><div><p><span>"Big Data processes codify the past. They do not invent the future. Doing that requires moral imagination, and that's something only humans can provide." </span></p><p>So, for work, I read a lot about big data. A lot. And it's all basically *jackoff motion* uhhnnnggg big dataaaa unnnnnnngggghhh yeah. And that makes me want to die.</p><p>This book is refreshingly critical of big data and algorithms, from a blessedly human approach. You might expect a lot of statistics and dryness, but it's a lot of real life stories to illustrate O'Neil's arguments, which makes it easy to read and care about. It's also a fairly quick read and doesn't waste a lot of time defining concept. O'Neil gets right to it, and her arguments are powerful. If I ever wonder why I can't seem to get my credit card APR down to a reasonable level even though my credit score is excellent, I have to think - maybe it's my zip code! So many pieces of data are taken into account for so many things that end up affecting poor people and people of color in wildly disproportionate ways. You might pay 5x more for auto insurance because you drive through "bad" zip codes at certain times of the day. </p><p>This book is pretty amazing and I'm so, so glad I read it. </p><p></p></div><section><ul><span><a href="https://www.goodreads.com/review/list/1459884?shelf=nonfiction"><span>nonfiction</span></a></span><span></span></ul></section></section></article></div></div></div>||||I||||        * Home
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              Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy

                Cathy O'Neil

                  3.88
                  27,679 ratings 3,369 reviews
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                Goodreads Choice Award
              Nominee for Best Science & Technology (2016)
                      We live in the age of the algorithm. Increasingly, the decisions that affect our lives--where we go to school, whether we can get a job or a loan, how much we pay for health insurance--are being made not by humans, but by machines. In theory, this should lead to greater fairness: Everyone is judged according to the same rules.
                      But as mathematician and data scientist Cathy O'Neil reveals, the mathematical models being used today are unregulated and uncontestable, even when they're wrong. Most troubling, they reinforce discrimination--propping up the lucky, punishing the downtrodden, and undermining our democracy in the process.
                  Genres Nonfiction Science Technology Politics Mathematics Business Economics
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                  259 pages, Hardcover

                  First published September 6, 2016

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                About the author

                          Cathy O'Neil

                          8 books 499 followers
                      Cathy O’Neil is the author of the bestselling Weapons of Math Destruction, which won the Euler Book Prize and was longlisted for the National Book Award. She received her PhD in mathematics from Harvard and has worked in finance, tech, and academia. She launched the Lede Program for data journalism at Columbia University and recently founded ORCAA, an algorithmic auditing company. O’Neil is a regular contributor to Bloomberg View.

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                        Arshias
                        28 reviews 24 followers
                  February 10, 2017
                      This was such a Malcolm Gladwell take on data science. I think this book touches on an important subject, and people should be aware of the issues O'Neil discusses. But instead of doing a deep dive into the subject, it just felt like a list of bad algorithms with instances of the people they hurt. It didn't contain many examples of "WMDs" that I had not already heard of, and that might be because she cited the New York Times for *like* all her case studies.

                      As someone who works in the field, I don't think this book was geared towards me. It wasn't technical or specific enough. Knowing her background, I really wanted her to get into the nitty gritty of some of the mathematics behind the algorithms. I know she is capable of doing this, but I think she instead chose to appeal to a wider audience. That's cool... except I don't think she did a great job of that either because this book lacked the context necessary to give people unfamiliar with the field a view of how machine learning and analytics typically work.

                      Ultimately, I thought the book felt unfocused, and it showed in the conclusion where proposed a series of pretty ridiculous recommendations within the span of about 15 pages. She has strong opinions on the topics she covered in the book. Although I agree with almost all of them, that isn't the point. The book is supposed to be about the algorithms, and instead she takes us on a tour of a collection of business and public policy malpractice, stating that the solution is to "encode values into our algorithms." Wut. Leaving the logistics of this aside... if the people doing the coding don't share your values (and they clearly don't), why would they do that? O'Neil herself noted a misalignment of incentives in many cases, particularly where the data work has been contracted out to other parties. Telling them to have values and trash their contracts is obviously not gonna fly! You can't fear-monger the whole book and then think everything will be happy sunshine rainbows once we take the data science Hippocratic oath.

                      I don't think there are easy answers, and I think it is ok to admit that. The takeaway should have been that data science isn't better than humans... because it is humans. We made the algorithms, so they run the gamut of use cases and demonstrate all the shades of gray we exhibit ourselves (yes, all fifty of them.)

                      TL;DR - Boo. I expected more from you, O'Neil.
                      dfa-book-club owned
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                        Yun
                        562 reviews 28.7k followers
                  February 28, 2022
                      I'm a data person. I pride myself on being logical and looking at the numbers before making decisions. And for quite a few years, I worked at a data visualization company and was a self-professed data geek. But can more data actually lead to worse results? That is what Weapons of Math Destruction tries to understand.

                      Insightful and timely, this book provides a detailed look at how algorithms based on big data don't always tell the truth or lead to a more fair world as they are purported to do. Rather, they contribute to a system that is opaque and hard to challenge, increasing the divide between the privileged and everyone else.

                      Each chapter provides a thoughtful exploration of an area where big data is supposed to be helping, such as college rankings, recidivism of convicts, applying for jobs, and getting loans. Algorithms in each area help define which are the best colleges, which convicts are most likely to reoffend, what personality types are best suited for a job, and who should get the best interest rates. That sounds useful, right?

                      But unfortunately, ideal data is not always available, so bad or irrelevant data is often used instead. And the resulting predictions are treated as gospel, increasing efficiency of the system, but harming those caught on the wrong side. It hurts a segment of the population while providing the rest of us with the false belief that fairness and justice has being done. In many cases, the algorithms' predictions create a negative feedback loop, directly influencing the outcome they were objectively trying to determine.

                      I found this book to be interesting and relevant. It really goes to show that your predictions are only as good as the data you've got. Whether you're a data geek like me or you just want to learn a little more about big data's potentially harmful effects, this is a worthwhile book to check out.
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                        Trish
                        1,386 reviews 2,646 followers
                  April 16, 2017
                      O’Neil deserves some credit right off the bat for not waiting until her retirement from the hedge fund where she worked to tell us the secrets of how corporations use big data ( our data). Underlying the collection and use of big data is an attempt to utilize efficiencies in the market place for goods, money, and talent. Big data ostensibly can also “set us free” from time constraints and uneven knowledge dispersal. Conversely the opposite is often true. We are at the mercy of how our own data is shredded and packaged, and errors in the model can mean mutually assured destruction—for the school, corporation, family.

                      The book starts with examples any readers who actually picked up this book to read might recognize: the chances of getting into a major university. O’Neil doesn’t go into the actual algorithms but just explains the variables chosen to populate the algorithms. Just when I was wondering who this book is targeted at, since after all, we kind of know how to get into university already, she comes up with examples of big data messing with aspirations that are still (hopefully not) in our futures.

                      She addresses the real pain-in-the-ass nature of minimum wage jobs where the inadequate part-time hours are constantly changing to maximize profits for owners and to screw with employees ability to plan their life, their children’s lives, and the children’s caretaker’s lives. O’Neil addressed the situation in 2009 when Amex decided to reduce the risk of credit card nonpayment by reducing the credit ceilings on users who shopped at certain stores, like Walmart. She shows us the way micro-targeting ends up using data to perpetuate inequities in opportunity and “social capital.”

                      The hardest part of reading this book (there is no actual math), was keeping my mind on what O’Neil was saying. Every time she'd mention another example of the ways big data was screwing us over, my mind would wander to experiences of my own, or ones I’d heard from friends, family, or others. This is real stuff, and just when I thought that it would be an excellent book for those with skills and interest in social justice to take to an interview with Google, Amazon, or a big bank, in she comes with another example of how the “fixes” are almost worse than the disease (Facebook’s method of who your friends are determining your credit risk).

                      But O’Neil reminds us big data, mathematics, algorithms, etc. aren’t going to go away.
                      "Data is not going away. Nor are computers—much less mathematics. Predictive models are, increasingly, the tools we will be relying on to run our institutions, deploy our resources, and manage our lives. But as I’ve tried to show throughout this book, these models are constructed not just from data but from the choices we make about which data to pay attention to—and which to leave out. Those choices are not just about logistics, profits, and efficiency. They are fundamentally moral."
                      Exactly. We still have to use our brains, not just our computers. It is critical that we inject morality into the process or it will always be fundamentally unfair in some way or another, especially if the intent is to increase profits for one entity at the expense of another. One simply can’t include enough variables or specifics. Some universities have begun to audit the algorithms—like Princeton’s Transparency and Accountability Project—by masquerading as people of differing backgrounds and seeing what kind of treatment these individuals receive from online marketers.

                      O’Neil suggests that sometimes data might be used to good effect by targeting frustrated online commenters with solutions to their issues: i.e., affordable housing info, or by searching out possible areas of workplace or child abuse and targeting that area with resources. She wades into national election data and notes that only swing states get candidates attention, suggesting, by the way, that the electoral college has outlived its usefulness to the citizenry. Algorithms are not going to administer justice or democracy unless we find a way to use them as a tool to root out inequities and try to find ways to deliver needed services where they are deficient.

                      When I look at the totality of what O’Neil has discussed, I am inclined to think this book is best targeted to thoughtful high schoolers and college-aged students who are thinking about planning their careers, who have a penchant for mathematical and computer modeling, and who think their dream job might be with an online giant. I’d be happy to be disabused of this notion if someone wants to challenge my thought that much of this information is known to many of us who have been out of school for awhile and who have been paying attention to our online experiences and junk mail solicitations. But it is always interesting to read someone as coherent and on the side of social justice as Ms. O’Neil.

                      It might be noted that Jaron Lanier in Who Owns the Future? (2013) also talks about the use of big data to steer our thinking and makes a preliminary suggestion that individuals should be paid for their data—for data that is collected about them, for profit. It is an interesting discussion as well. Love these intersections of technology and humanity.
                      business computerology economics
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                        Leo Walsh
                        Author 3 books 123 followers
                  April 5, 2017
                      Captivating. Insightful. And important. A 50,000 foot view of how automated big-data is a great tool for understanding human nature. How it has great promise to make our lives easy. And yet, a very real takedown of how systems engineers -- and corrupt power-seekers, like corporate executives and for-profit universities -- misuse this powerful tool. And the even worse cases where people start with good intentions, like ridding school systems of bad teachers, only to toss out "false negatives."

                      I found Cathy O'Neil's "Weapons of Math Destruction" a very important book that highlights a lot of what's been going on in America over the past 30 or 40 years. For instance, I just finished "The New Jim Crow," and wondered how the Supreme Court would continue to rule in favor of crime policing tactics that 1) target poor urban areas -- populated by mostly black men, and 2) allow the police the ability to stop people, willy-nilly, that they found "suspicious" -- despite the fact that minorities are caught in stop-and-frisk situations while most are innocently going about their lives. Problem is, many of these people going to jail for nuisance crimes -- possessing small amounts of marijuana, open containers, driving on expired tags, etc. Things that seem the exact definition of "systematic racism."

                      But when O'Neil lays out how systems engineers have written algorithms that send police to "hot spots," those rulings make sense. For instance, large crimes that we all want to stop -- car thefts, burglaries, rapes, assault and murder -- are rare. While petty and nuisance crimes -- jaywalking, possessing weed, noise violations, vandalism, etc. -- common. Based on the law of large numbers, a program trying to optimize "broken windows" policing would send more officers to a district with a higher concentration of people, typically ghettos. Which leads to more petty arrests in those areas for crap that doesn't matter much. Which sends even more police into these areas, making more arrests for petty offenses... and so on and so on.

                      In short, these ghettos are caught in a negative feedback loop. And the residents more likely to end up in jail for something stupid. Like smoking dope which, based on most evidence, is just as high among whites than blacks. So a white 19-year-old frat boy at Ohio State can smoke-up at will, with little possibility of being caught. While a nearby ghetto-dweller, who works maintenance or in the office at the university, or attends the university while living at home, will have a greater likelihood of being arrested. Just based on where they live.

                      Same crime -- use of narcotics. Two different outcomes.

                      That is what O'Neill terms as a "weapon of math destruction." Since it is pervasive, destructive, and opaque.

                      And things get worse if the two hypothetical dope-smokers get arrested. Odds are, the white frat boy's parents live in a safe suburban neighborhood and thus knows zero convicted felons. So his court-appointed "recidivism" score -- attained by another destructive math weapon -- will be lower than the ghetto dweller's who lives near many felons. Thus, the courts may lessen the frat boys charge to a misdemeanor while charging the ghetto dweller with a felony. Due to that recidivism score.

                      Talk about kicking someone when they're down.

                      And when both released on parole, and ordered to "steer clear of felons." the frat boy will have no problems following this. While the ghetto kid, thanks in part to the policing software noted above, will be surrounded by them. Which, of course, adds to the already disadvantaged the further risk of being pegged as a parole violator.

                      Smack.

                      O'Neill lays out other ways that system engineers perpetuate injustice. She takes-down for-profit universities -- like the University of Phoenix -- who actively target poor people with aspirations. Not to help, since a University of Phoenix degree costs tens of times more than a community college degree, while adding less salary. Instead, University of Phoenix and their ilk exist to cash in the student loans guaranteed by the government.

                      Yep. Poor people, who don't know any better, are targeted by companies that give them less while charging them more. Using data-driven web advertising, another Weapon of Math Destruction, all for a quick buck.

                      These are just a few WMD's that O'Neill examines. She looks at others credit scoring, e-scoring (a sort of electronic "credit score" derived on you based on your social media friends and activities), the USA Today college rankings, which have lead to spiraling tuition costs while providing questionable value.

                      All of these WMD's lead to increasing social stratification. And, in many ways, drive the "winner take all" nonsense that gives big money to a handful of developers who program an app.

                      But the nice thing is that O'Neill ends by providing valuable insight into how, when properly used, deep-dive statistics can actually help people. For instance, O'Neill was part of a task-force that examined New York City's homeless. They uncovered the single unequivocal variable that would keep people off the streets -- access to Section 8 housing. And once these people were housed, they'd move on to get jobs --since having a stable residence makes it easier to gain employment. And thus, less likely to end up on the streets. And all this research came at a time when Mayor Bloomberg was contemplating reducing Section 8 housing. So it's prove important.

                      She also points to other positive uses of algorithms -- all of which point to putting our compassionate, human-based morality ahead of the appearance of objective, measurable efficiency -- with "appearance" being the operative word. Since O'Neill makes the cost of following these algorithms clear.

                      All-in-all, "Weapons of Math Destruction" is the best science book I've read over 2016. Since it focuses not only what we can do -- the science and how it makes things more efficient -- but forces us to focus on the ethics, the "why" we may choose a less efficient alternative as it may be more just. Especially when blindly accepting a model often degenerate to "pseudoscience." And that anti-scientific narrative can be amplified if the person wielding the WMD is either greed or malicious.

                      5-stars.

                      That said... YAY!!! This is my 80th book of the year. So I've just completed my goal.
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                        Clif Hostetler
                        1,173 reviews 878 followers
                  January 18, 2019
                      "Welcome to the dark side of big data." Thus the author concludes the Introduction section of this book. Computers and the internet have enabled us to advance into the new world of algorithms and big data with ramifications that most people are unaware of.

                      Surfing the web, clicking "like" in Facebook, Googling (i.e.searching on line), and making online purchases are common examples where big data is either tracking and potentially impacting our lives. Some of these are benign and can be helpful. But this book focuses sharply in the other direction, on the "harmful examples that effect people at critical life moments, going to college, borrowing money, getting sentenced to prison, finding and holding a job. All these life domains are increasingly controlled by secret models wielding arbitrary punishments."

                      One of the most shocking pieces of information provided by this book are the examples of how big data contributes to economic inequality. It is devastating how efficiently private colleges and payday loan companies target the economically stressed portion of the population. It's also astounding how frequently limitations and flaws of big data are ignored and substituted for the truth. It's almost comical how big data systems can be manipulated (a.k.a gaming the system) by clever institutions and companies.

                      Whether we like it or not this is the new world in which we all live. Citizens of this new environment need to become knowledgeable about how big data works. Otherwise we will all be its clueless victims.

                      The following are some quotations from the book.

                      The following are the closing sentences of the Introduction:
                      Big data has plenty of evangelists, but I'm not one of them. This book will focus sharply in the other direction, on the damage inflicted by WMDs and the injustice they perpetuate. We will explore harmful examples that effect people at critical life moments, going to college, borrowing money, getting sentenced to prison, finding and holding a job. All these life domains are increasingly controlled by secret models wielding arbitrary punishments. Welcome to the dark side of big data.
                      In the following the author defines some of the shortcomings she observed in how big data was being used.
                      More and more I was worried about the separation between technical models and real people and about the moral repercussions of that separation. In fact I saw the same pattern emerging that I had witnessed in finance, a false sense of security was leading to widespread use of imperfect models, self-serving definitions of success, and growing feedback loops. Those who objected were regarded as nostalgic Luddites.
                      The author worked at a hedge fund during the 2008 financial collapse. Thus when she moved into the field of consumer data modeling she looked for flaws in the use of data that were similar to what led to the credit crisis.
                      I wondered what the analog to the credit crisis might be in big data. Instead of a bust I saw a growing dystopia with inequality rising. The algorithms would make sure that those deemed losers would remain that way. A lucky minority would gain evermore control over the data economy raking in outrageous fortunes and convincing themselves all the while that they deserved it.
                      The following is the author's summary near the end of the book.
                      In this march through a virtual lifetime we've visited school and college, courts and the work place, even the voting booth. Along the way we have witnessed the destruction caused by WMDs. Promising efficiency and fairness, they distort higher education, drive up debt, spur mass incarceration, pummel the poor at nearly every juncture, and undermine democracy.
                      The author, Cathy O'Neil is highly qualified (she got her Ph.D. in math at Harvard), had work experience from inside the system (a “quant” at D.E. Shaw—a major hedge fund), and has evolved into a "Occupy Wall Street" activist. Thus she has experience and qualifications needed to explain how big data effects our lives. It's needed information.

                      Author's Ted Talk:
                      https://embed.ted.com/talks/cathy_o_n...

                      Cathy O'Neil was the winner of the 2019 MAA Eurler Book Prize:
                      https://www.maa.org/programs-and-comm...
                      current-events
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                        Trevor
                        1,362 reviews 23.1k followers
                  August 26, 2018
                      We like to think of mathematics as basically pure and free from the nastier side-effects of human nature. And this purity rubs off – so that the closer a science is to being able to be described in numbers, the more highly we regard it – so, physic is seen as somehow higher than biology, and economics than anthropology. That is, if you can predict behaviour on the basis of an algorithm – whether that be the behaviour of a billiard ball or a homeless person – then this is proper science and it has a claim to a kind of objective truth that sets it aside from being challenged.

                      And that is the point of this book – it seeks to help shatter this illusion, particularly in relation to the human sciences, but also, and perhaps more importantly, in relation to marketing, insurance, policing, education and other social activities that are increasingly being modelled and even normalised by algorithms. She refers to these algorithms as the weapons of math destruction in the title (the title is better in English, of course, where we say maths, rather than math, but her point stands).

                      The destruction such algorithms can cause became clear to her while she was working in finance just before the 2008 crash. It certainly isn’t that she sees maths itself as being the problem, she refers to herself almost immediately as a kind of math nerd and proud of that designation. Her point is that these algorithms are dangerous because we tend to think of them as being purely objective and therefore the results they provide as being beyond question. And this state is helped along by more than just the fact we hold mathematics in such high regard. It is furthered by the fact that often the algorithms that spit out these assessments of us are obscure, unfair and grow exponentially. These are the three conditions that the author believes makes an algorithm a likely WMD.

                      So, in looking at these in turn, is the algorithm is obscure? – and most of them tend to be, as she says at one point, they are the ‘special sauce’ for many companies and so they need to remain hidden from the competition. For instance, if you have an algorithm that allows you to predict who is going to make an ideal partner for someone else, then your dating site is going to make you lots of money. You are hardly going to want to let your competition know you secret. But the problem is that by keeping your algorithm obscure and hidden from outside analysis, you can say nearly anything you like about its effectiveness, if no one is then able to check. You know, this is a version of the ‘they can vote any way they like, as long as I get to count’ story.

                      But it isn’t just outright fraud that is the problem here – although, that doesn’t mean fraud isn’t a problem. Just as bad is the idea that often these models bury their false negatives. That is, if the algorithm says ‘never employ anyone with green eyes, they are under-performers’ the company that follows this advice is unlikely to ever find out if this is true or not. That’s because they won’t have employed anyone with green eyes to test the model – so the model will be confirmed by default. The problem is that many of the algorithms used – say in policing – can encourage over-policing of certain populations (this is set in the US, so let’s just call those certain black, Hispanic and Muslim populations to save time) and this over-policing, by defining certain populations in these ways are also likely to create the monster they were supposed to be eliminating. A nice example is an algorithm that denies jobs to people according to their credit scores, which then means these people are less able to pay off their debts, which gives them a worse credit score, which then confirms the risk – vicious cycle anyone?

                      Other examples focus on the use of psychometric tests for all manner of things, but increasingly as a pre-employment test. You might think these ought to be relatively easy to game – you know, how stupid would you have to be to answer ‘very true’ to the prompt, ‘I sometimes fly off the handle for no real reason’. You know, unless you are going for a job in as a wrestler, your employer is probably not going to be overly impressed with that answer. But as she points out, sometimes applicants, for example, for a job at McDonalds, are asked to choose between one of two alternatives: “It is difficult to be cheerful when there are many problems to take care of – or – Sometimes, I need a push to get started on my work”. I have no idea what the ‘right’ answer to that choice is. I’m not even sure which of the two really applies to me – although, there have been times in my life when both of them have, and in your life too, I suspect. That is, it isn’t at all clear what this is seeking to achieve, but it is clear that there is likely to be a ‘wrong’ answer, in the sense that making that ‘choice’ might leave you without a job.

                      But even this level of obscurity isn’t her main worry (at least you know you’ve been asked a question here that might be used against you) – but it is too often easy to correlate factors to ensure that certain people are excluded due to their sex, sexuality, race, social class and so on, merely by the underlying assumptions of those programming these algorithms – and the algorithms aren’t ‘objective’ examples of the purity of mathematics, but rather socially produced ephemera that are likely to have been shaped by the social stereotypes of the society and the people who produce them.

                      The author mentions that curious fact that orchestras now appoint five times as many female players since auditions have been held with the player behind a curtain. Who’d have thought women would play so much better when they were hidden from sight? Such shy and retiring little things, bless them…

                      The solution is to ensure that algorithms used to judge us are open and available for anyone to check – a condition that will become increasingly unlikely as these algorithms are proprietary.

                      The question of fairness isn’t at all easy to address and for many of the reasons already mentioned. One of the examples given is teacher scores being used to determine who should get pay increases and who should be removed from the profession. Basically, the idea is to compare student outcomes so that the teachers who do not increase student scores enough should be shown the door. But, as is pointed out here, student scores aren’t only affected by teacher performance. And worse, if you are teaching students who are either very far behind or very far in front you are unlikely to affect as big a movement in their scores as you might if you are teaching kids in the middle. If you are going to be assessed on the basis of a score, that score ought to be related to something you have control over, rather than merely something that is relatively easy to measure. The author points out that far too often the kinds of assessments made of teachers based on student attainment produce virtually random results year on year. Since it is very unlikely that an exceptional teacher will become a very poor teacher from one year to the next, then any assessment that produces such a result ought to be suspect. And this is also true of loan application algorithms that judge you based upon the people you associate with. For instance, I’ve read a few times lately that your credit rating can be impacted by your ‘friends’ on FaceBook, but even if this is not literally true, algorithms are shown here to judge you by many factors that assume associations between you and other people more or less likely to be a credit risk. For instance, if you buy things in certain stores, as American Express admitted recently they restricted access to credit to people based on them frequenting particular stores.

                      The last condition of a WMD is that the algorithm can be scaled up to cover large populations. An algorithm that works well in one set of circumstances might be both transparent and fair within its limited application, but because it is then being used across a larger population it might suddenly stop being fair or reasonable. This is because it creates a new norm in the population at large and that might well have seriously disadvantageous impacts on populations beyond the one it was originally intended to be used upon. Again, the examples I jump to tend to be associated with education, where large scale testing programs have a disproportionate impact on poor communities which are defined as failing and then ‘success’ becomes defined as ‘doing well on the test’ – so that the tail starts wagging the dog. And this then has impacts on how kids get taught – if you are only going to be measured by test results, then we should just prime you for taking tests. Which then makes education as boring as it is possible to make it for kids that were already struggling to see the point of education in the first place. But the communities that do well on these tests are normally the already advantaged and they suffer none of these negative impacts, because, well, they do well on these tests, so no point priming them on more of them…Let them do art.

                      I really would recommend this book – she gives lots and lots of examples and it is vitally important that we understand that this is the world we are increasingly moving towards. More and more of our lives are going to be influenced by algorithms and big data – and yet too many of us are so terrified of mathematics that we will blame ourselves when these algorithms punish us. It isn’t at all clear to me how we might go about making these algorithms transparent, fair or limited to a scale that keeps them safe – but these are questions we really ought to think about and act upon.
                      economics education maths
                        ...more
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                        Nilesh Jasani
                        1,083 reviews 204 followers
                  November 27, 2016
                      For the most part, WMD is a rant with only impractical statements as solutions.

                      The author is onto something critically important when one reads the title of the book and goes through the first few pages. However, it is a tragedy to see the author falling in love with her own phrase WMD and completely losing the plot. The examples used are good in the beginning but soon turn ridiculous (they would be laughable if not so lamentable for the people involved). In the process, the author loses her credibility as a champion of the topic.

                      Consider the following example: a person is unable to find a job as he claims he fails all pre-interview personality tests. The author debunks the objectivity of tests in favor of the subjectivity of interviewers with the claim that in another world, one or the other interviewer would have given the job to this person. The author never thinks about the interviewers' non-transparent and more difficult to fathom biases wreaking greater damage in that world versus the current one where the biases are at worst in models that are far easier to scrutinize and amend. The author failts to realize that every world with fewer desirable jobs compared to applicants will leave some unhappy - the fairness is perhaps far better where the selection process is more equal for all.

                      In a way, the author wishes "we" to stand up and help reduce the use of math. The author is clear in the "reduction" goal without ever realizing how "we" would ever decide to what extent the reduction should take place and whether the constituents of "we" could ever agree on the replacement. The repeated call for subjective biases assumes that the biases come in only one form and are acceptable to all. The absurdities reach a peak when the author declare that at least subjective biases evolve over time and hence should be preferred while the mathematical models are static!

                      The author's hatred for anything involving models not only creates a highly one-sided discourse but also leads to inadequate treatment of any important points within. The only mathematical concept discussed somewhat is that of the circularity caused by the self-reinforcing feedback in multi-equilibrium equations leading to solutions that might be locally optimum but not globally (all my words). Otherwise, the book almost exclusively focuses on the unsuitability of results arrived through statistical sampling on any individual case.

                      To summarize, the author seems to yearn for insurance companies to abandon actuarial methods, schools to move away from test scores for admissions and financial institutions/tech companies/governments to reduce their use of numbers. The author wants humans to take decisions that these models might be taking using their value judgments. Or the world to simply not take those decisions.

                      There is so much sensible that needs to be discussed on the damage created by the excessive use of math in the modern world but this book is not the best place to explore it.
                      non-fiction
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                        Mario the lone bookwolf
                        805 reviews 4,960 followers
                  December 11, 2018
                      Is it legitimate to reduce people to the data that can be extracted from them?

                      Please note that I put the original German text at the end of this review. Just if you might be interested.

                      Especially the predictions made possible by big data are a frightening aspect so that behavior and personal development can be predicted with increasing probability. Also, like any artificial intelligence, the algorithms and programs become more and more efficient, both in parallel with the growing amount of data about the individual and through the optimization of the functioning.
                      Whether habits, buying behavior, Facebook Likes, movement patterns, writing style, illnesses, search machine inputs, political activities, professional aspects or social contacts, everything flows together in a large memory.

                      Particularly explosive is the ability to accurately predict the electoral behavior of specific groups of the population and thereby be able to address election advertising even more targeted. It also opens up the option of manipulating groups, which are more likely to be responsible for the competition camp, utilizing disinformation and misinformation in their awareness-raising and decision-making. Even outside of elections, the data volumes offer promising perspectives for influencing opinions, facts, trends and mood barometers.

                      The subtle, digital prompters are hardly recognized as such. The information is automatically trimmed in the form of auto-completion and suggestion functions, the individually compiled search results and purchase recommendations based on the previous buying behavior. Moreover, directed to the individual interests or the opinion to be shaped. Even what you type in and then delete again or also revise and change, is saved.

                      This can be considered optimistic as practical, time-saving and relieve and helper at work. Alternatively, as an indirect kind of influencing, over which only the compulsory consumption is promoted as optimized as possible. As an alternative to the offer of tailor-made products, there is the option of deliberately focusing just on specific information and news. So that unpopular and annoying reports, opinions and activists are intentionally not even displayed in the higher result ranges. To merely skip the scissors in the head and defamation immediately and proceed directly to censorship.

                      Perfide is the playful way in which free offers lure the consumer. For example, a health application for smartphones, with which one competes in real time with friends, in which insurance and health insurance should also be interested in. These hands-on free programs provide a fun and competitive and user-driven option for businesses to access users' location, interest, and behavioral data.

                      The book is an excellent appetizer, with the benefit of awakened interest in possible deepening of the study of the topic by further and detailed, related literature.
                      Given the overall situation a worthwhile endeavor. Contrary to the apathy and disinterest of large populations owed the fact that it could come too far. So without long-awaited outcry across all layers of civil society, much more could go on.

                      In Western countries, social networks, search engines, and Internet merchants are only interested in manipulating buying behavior and collecting data to monetize it. The direction of China with the Citizen Score is likely to find imitators. Then, the arbitrary and negligently disclosed data would not anymore only decide the price of a product or the next banner ad. However, about your future and maybe your survival or death. For example, if you criticize the government.

                      Ist es legitim, Menschen auf die Daten zu reduzieren, die sich aus ihnen extrahieren lassen?

                      Gerade die durch Big Data möglich werdenden Prognosen stellen einen erschreckenden Aspekt dar, lässt sich Verhalten und persönliche Entwicklung auf diese Weise doch mit immer höherer Wahrscheinlichkeit voraussagen. Und wie jede künstliche Intelligenz werden die Algorithmen und Programme sowohl parallel zu den wachsenden Datenmengen über die jeweilige Person als auch durch die Optimierung der Funktionsweise immer effizienter. Ob Gewohnheiten, Kaufverhalten, Facebook Likes, Bewegungsmuster, Schreibstil, Krankheiten, Suchmaschineneingaben, politische Aktivitäten, berufliche Aspekte oder Sozialkontakte, alles fließt in einem großen Speicher zusammen.

                      Besonders brisant ist die Möglichkeit, Wahlverhalten bestimmter Bevölkerungsgruppen präzise vorhersagen zu können und dadurch Wahlwerbung noch zielgerichteter adressieren zu können. Es öffnet auch die Option, eher dem Konkurrenzlager anzurechnende Gruppen gezielt mittels Des- und Falschinformation in ihrer Bewusstseinsbildung und Entscheidungsfindung zu manipulieren. Auch abseits von Wahlen bieten die Datenmengen verheißungsvolle Perspektiven für die Beeinflussung von Meinungen, Fakten, Trends und Stimmungsbarometern.

                      Dabei werden die subtilen, digitalen Souffleure nur schwerlich als solche erkannt. In Form von Autovervöllständigungs- und Vorschlagfunktionen, den individuell zusammengestellten Suchergebnissen und auf das bisherige Kaufverhalten angelehnte Kaufempfehlungen werden die Informationen automatisch beschnitten. Und auf die individuellen Interessen oder die zu prägende Meinung hin gelenkt. Selbst was man eintippt und dann wieder löscht oder noch überarbeitet und verändert, wird gespeichert.
                      Das kann optimistisch als praktisch, zeitsparend und die Arbeit erleichternd betrachtet werden. Oder als eine indirekte Art der Beeinflussung, über die vorerst nur der Konsumzwang möglichst optimiert gefördert wird. Alternativ zur Offerierung der auf den Leib geschneiderten Produkte bietet sich die Option, gezielt nur bestimmte Informationen und Nachrichten in den Fokus kommen zu lassen. Damit missliebige und lästige Berichte, Meinungen und Aktivisten gezielt gar nicht erst bis in den höheren Ergebnisrängen angezeigt werden. Sondern die Schere im Kopf und Diffamierung gleich zu überspringen und direkt zur Zensur voranzuschreiten.

                      Perfide ist die spielerische Art und Weise, mit der der Konsument durch Gratisangebote geködert wird. Zum Beispiel eine Gesundheitsapplikation für Smartphones, mit der man in Echtzeit mit Freunden konkurriert, was Versicherungen und Krankenkassen auch interessieren dürfte. Diese praktischen kostenlosen Programme ermöglichen eine spielerische und auf Wettbewerb und den Ehrgeiz der Anwender zielende Option für Unternehmen, an Standort-, Interessens- und Verhaltensdaten der Benutzer zu kommen.

                      Das Buch ist ein guter Appetitmacher, mit dem Interesse an einer etwaigen Vertiefung der Beschäftigung mit dem Thema durch weiterführende und auf einzelne Aspekte detaillierte eingehende Literatur geweckt wird. Angesichts der Gesamtsituation ein erstrebenswertes Unterfangen. Entgegen der, der Apathie und Desinteresse breiter Bevölkerungsgruppen geschuldeten Tatsache, dass es so weit kommen konnte.Und, ohne längst fälligen quer durch alle Schichten der Zivilgesellschaft gehenden Aufschrei, noch viel weiter gehen könnte.

                      In den westlichen Ländern sind die sozialen Netzwerke, Suchmaschinen und Internethändler nur an der Manipulation des Kaufverhaltens und dem Sammeln von Daten zu dessen Monetarisierung interessiert. Die Richtung, die etwa China mit dem Citizen Score geht, dürfte Nachahmer finden. Dann entscheiden die willkürlich und fahrlässig preisgegebenen Daten nicht nur mehr über den Preis eines Produkts oder das nächste Werbebanner. Sondern über die eigene Zukunft und vielleicht auch das eigene Überleben oder Sterben. Wenn man etwa die Regierung kritisiert.
                      0-internet 0-social-criticism 0-technology
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                        Suzanne
                        1,726 reviews
                  September 7, 2016
                      This book did a nice job describing large-scale data modeling and its pitfalls in a very accessible manner. It is so easy to think of computer algorithms as unbiased; however, the author demonstrates how they really do discriminate. Next time I teach a class involving statistics, I may use this book to show students how it is dangerous to blindly believe the numbers.
                      netgalley nonfiction
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                        David Rubenstein
                        828 reviews 2,690 followers
                  December 10, 2017
                      The subtitle of this book, How Big Data Increases Inequality and Threatens Democracy really says it all. Big data has come into our lives in numerous ways, and many of them are a scourge on our lives. Big data, in and of itself, is not to blame, but the uses to which it is put are often outrageous. Take the case of automated teacher evaluations. These are often based on the improvement of students' scores. It seems like a no-brainer, and since the scores take into account the improvement rather than the absolute scores, they seem to be very fair. However, one New York teacher received an abysmal score of 6 (out of 100) one year, and the following year received a wonderful score of 96. Obviously the teacher did not suddenly improve his teaching methods.

                      If a mathematician were to analyze the scores from such evaluations, their random scatter would be instantly recognized as meaningless. Yet, these automated evaluations are used for hiring/firing decisions, as well as compensation decisions. The worst aspect, is that there is no attempt to improve the algorithms. The algorithms used in these evaluations are opaque, and no attempt is made to apply feedback to tweak the scores to make them more accurate and fair.

                      Such algorithms are used in many avenues of life. Credit scores are used by loan companies--and even by auto insurance companies! Police departments use algorithms to plan targeting of police activity, while courts use algorithms to predict recidivism. Colleges, especially for-profit colleges, use algorithms to target potential students.

                      While this book comes across as very preachy, and very liberal, it does make some very good points. Big data is often used--intentionally or not--to punish the down-trodden and to increase inequality. Profitability is usually the goal, and while fairness is often the purported goal, these algorithms rarely turn out to have the effect of being fair.

                      I recommend this book to people who are curious about the effects of big data on their lives and on democracy.
                      mathematics politics
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                        Emily
                        735 reviews 2,444 followers
                  February 7, 2017
                      Big Data is opaque, complicated, managed by profit-seeking corporations, and is more and more dictating certain societal conditions: from getting a job to applying to college to receiving healthcare. "Data," on its own, seems amoral, a way to implement systems that are more fair. But O'Neil's point in this book is that all algorithms include basic assumptions, and sometimes those basic assumptions are full of bias and not grounded in fact. If the algorithms aren't regularly inspected, they create their own feedback loops. As O'Neil says, "In each case, we must ask not only who designed the model but also what that person or company is trying to accomplish."

                      This book touches lightly on a number of topics to give an overview of the algorithms that O'Neil finds the most objectionable. Each of the different sections is interesting, but they don't delve particularly deeply into the topic; for example, if you're interested in how poor feedback loops on policing and prisoner recidivism work together to create our prison industrial complex, you're better off reading The New Jim Crow . But if you're interested in an introduction to the ways in which data runs our lives, this is a good place to start.

                      I was particularly interested in the section on college admissions, because O'Neil ties skyrocketing college tuition prices to the US World News ranking (the first ranking for universities in the US). In 1988, journalists used a number of proxies to create a way to rank colleges against each other. This example hits on a few of the points that O'Neil brings up again and again: the journalists wanted proxies that would reaffirm their own biases, so Stanford and Harvard had to be at the top of the list. But they weren't able to fully factor in the things that mattered, like the quality of education, so they relied on proxies like admissions rate and freshman retention rate. Cost was conspicuously excluded, and thus - or so O'Neil says - college tuition has increased 500 percent between 1985 and 2013. Colleges are gaming the system to get themselves back up to the top of the list, but then are passing those costs directly on to the students who attend.

                      While this book is nothing groundbreaking, it's another reminder that tech HAS to do better if we are going to put more of our personal data - our lives - into its hands. I personally believe we're all being swindled by the "sharing economy" (heavy quotes there), and the example about Lending Club hammers that home. O'Neil uses Lending Club as an example of a service that was supposed to "democratize" loans, but it's quickly devolved into 80% institutional money: that is, money from big banks. Why? Banking has regulations, and banks aren't legally allowed to discriminate against consumers on the basis of e-scores and other algorithmically created scoring systems that take into account factors like zip code, punctuation on applications, and social networks. These institutions have found a way to get outside of the law, apply their sketchily drawn "data" models (that have no statistical basis), and reap the rewards, while further exacerbating class inequality.
                      politics science
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                        Scott
                        34 reviews 4 followers
                  October 25, 2016
                      Book reviews are all about expectations, and honestly I, as someone doing data science and grappling with issues, expected more. With a data scientist writing a full length book inditing data science one expects a deep dive revealing real points. Instead it ends up being a very surface level essay without the deeper exploration and meaning one expects from a full length work. Perhaps more worrisomely, her own definition of a WMD it introduces is often worked around to bring in arguments she wants to make.

                      I do agree with her main argument/definition of a Weapon of Math Destruction(WMD), though the name is far too punny for real use. More or less the author argues that bad algorithms are secret, and expand out rapidly without checks on actual results. This is hardly a novel problem statement, but it is certainly a real problem. However several times she applies shortcuts of assumed results to actual results in evaluating presumed WMDs which is the very same problem she is complaining about. She also seems to have added an unspoken fourth criteria being anything that offends her liberal (and she is very liberal, and interventionist) ideals. This fourth criteria is apparently enough to give a pass to what are clearly bad algorithms that fail her own tests. This is needless to say, not a great way to prevent issues with unethical use of algorithms, unless you happen to have the same ideals and goals as Cathy O'Neil.

                      Somewhat surprisingly, this is a data science book without any data. It doesn't even have in text footnote (at least in my kindle edition, though they do sometimes update that), and the notes section at the end shows the citations to be really quite poor and sparse. We are talking college freshman "The professor said 10 citations so I better cite a bunch of random webpages" level of citations. To be fair her arguments are only weakly data driven, but I still expected something with a deeper academic backbone.

                      This is also a data science book that won't teach you anything about data science. If you are looking for that you should look elsewhere. The author has co-written another book Doing Data Science , and it shows her quite capable of handling the topic. I do wish she had brought more of that into this book, which will be many people's first book in data science.

                      The structure of the book plays out more as a series of related blog posts than a book that builds on themes and has progression. Partly this may be an unfortunate result of the author being a blogger, but it leaves one wanting more.

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                        Maru Kun
                        220 reviews 522 followers
                  August 22, 2017
                      Forget those cute pastel illustrations from the fifties with their flying cars, robot servants and dreams of unlimited leisure. Our future has finally arrived and for most of us, especially for the less rich and less privileged who won’t qualify for individualized attention, the computer says “No”.

                      ‘Weapons of Math Destruction’ is a timely book about the increasing influence of algorithms to control the news we see, the jobs we can get and the politicians we vote for; algorithms working tirelessly on someone’s behalf (not yours), unseen and unaccountable.

                      The book explains statistical and methodological problems with these algorithms and illustrates how these same problems manifest themselves when they are applied to real world situations.

                      An example gives a flavor of the issues that recur throughout the book:

                      An algorithm which compared changes in student performance year-on-year was used to decide which teachers were a poor performers who would be let go. In one terminated teacher’s case subsequent evidence suggested that student’s scores had been tampered with the previous year and as a result she had inherited a class whose performance had been overstated, so was bound to deteriorate with the algorithm marking her out as a poor teacher.

                      The teacher was not told how the algorithm applied nor allowed to appeal the decision; the algorithm was a black box that produced a result the school system wanted – terminations. And, notwithstanding the lack of transparency in this decision, a sample of scores from one class alone - perhaps twenty to thirty students - was not sufficient to produce a statistically valid result in any event. One more reason why being an American public school teacher must be one of the worst teaching jobs in the developed world.

                      Please read the book for more egregious examples of “algorithm abuse”. Below is my summary are some of the key problems the book outlines. It is helpful to think about these problems in two categories, problems which come from poor application of the technical aspects of the algorithms (bad programming, misapplication of statistical or machine learning methods) and problems which come from how the algorithms are developed and used, how they incorporate hidden biases or value judgments, poorly thought out objectives and other ‘human factors’ beyond just poor maths:

                      Some of the common issues were:

                       Lack of transparency in how the models operate and how they make a decision, often leading to no means of appeal against a clearly unjust outcome. Related problems include over reliance on models in the face of contradictory data or, where people do understand the models but are benefitting from them, a lack of integrity in applying them. Models used to price mortgage backed securities during the financial crisis are a leading example.

                       The use of “proxy data” for model input or output because the real data desired is unavailable, too expensive to obtain or cannot be objectively measured. Does sending out more e-mails with “creative phrases” mean that you are really a more creative and innovative person? FICO scores are a relatively good model for use in predicting credit risk, but not as a proxy when used for a whole host of other unrelated things such as predicting future job performance when used in the hiring process.

                       Feedback loops, whereby a model increasingly encourages non optimal outcomes by rewarding certain behaviors at the cost of intended benefits. An example given is US college rankings which increasingly reward “user experience” and “research citations” rather than the actual educational outcomes for students.

                       Algorithms that, mainly for efficiency purposes, use data from arbitrarily selected groups when individual data is available. Why should an algorithm price insurance for a driver based on the experience of other drivers that live near him or are in a similar economic position instead of on his own individual driving record?

                       Optimization that is good for an algorithm’s owners but not for society as a whole. Monetary return is the most common priority for an algorithm used in the private sector, but is this what society wants when it results in micro-targeting the poor with the marketing of for-profit colleges that provide a below average education at great cost?

                       Hidden biases and unfairness when assumptions are built into an algorithm that are reflective of social factors rather than individual experience. Algorithms used by the police to forecast crime are first used to predict easy to identify nuisance crimes which, unsurprisingly, occur mainly in deprived neighborhoods. The police have yet to develop an algorithm that forecasts where white collar crime takes place, although if they did Wall Street would surely light up red.

                       Incorrect use of statistics; lack of feedback of model results into predicting outcomes. Baseball is a good field for prediction because outcomes – homes runs, strikes, batting averages - can be objectively measured and predictions can be fed back into the model to improve it. Using FICO scores in recruitment is not what the developers of FICO intended; there is no study suggesting statistical correlation between FICO and subsquent good job performance; good or bad job performance is also never used to assess HR models or improve their reliability.

                      The last chapter of the book looks at the micro targetting of voters with political messages through Facebook and similar social media sites. This was written in September 2016, a respectable amount of time before the world had a glimpse into the dizzying vortex of fake news, Russian hacking and tweeting Presidents. This is an excellent chapter and may well be seen as prophetic once we look back over our current period of political chaos, if it ever ends.



                      I resolved to write notes on the books I read as I get so much more out of them if I do. This resolution will last about one and a half books I’m sure, but if anyone is interested here they are:

                      Introduction: Mathematical models are being increasingly used to make decisions that have real world impacts. However the algorithms they use are opaque except to a limited number of mathematicians or computer scientists and may, sometimes unknowingly, encode human prejudices, misunderstandings and biases.

                      Assumptions may be camouflaged by the maths and go untested and cannot be questioned by the persons to whom they are applied. These algorithms are often used in inappropriate contexts where there is insufficient objective data to properly apply the underlying statistical theory; algorithms may work for baseball with a many tens if not hundreds of thousands of objective, constantly updated data points but not to evaluate a teacher with a class of thirty.

                      The algorithms produce “feedback loops” under whereby they produce output that eliminates particular classes of result based on false assumptions, but as a result reinforce the weighting given the assumption in the model. Injustice is reinforced as these algorithms are applied to ever larger numbers of people, although the rich and privileged may still be assessed on an individual basis rather than as one of the masses assessed by machine. People unfairly denied opportunities as a result of this software are “collateral damage”.

                      The algorithms optimize a payoff designed by the developers of the model - in the case of many private sector models, monetary profit - but who is to say that this is the optimum payoff for society as a whole?

                      Chapter 1, Bomb Parts - What is a Model? The author explains how her managing of her family cooking is a form of “data model”: inputs are family preferences, appetite that day; available food, special cases such as cooking on a birthday, the output is “family satisfaction” and the model determines menu for that day The model could be pre-programmed with a set of rules to determine its menu or could be trained through observing many examples. In either case mistakes may be made, perhaps through forgetting a rule or not including a rare case in the training data. The key point is that the model can incorporate personal biases that are not visible, in the case of the menu model towards healthy food and away from ice-cream.

                      Models don’t have to be complicated to be effective; a smoke alarm is a model intended to identify fire that operates on only a single input, the concentration of smoke particles.

                      Three questions are posited to evaluate models: Firstly, is the model understandable - or even visible to - by the people to whom it is applied? Secondly, does the model work in the subjects’ interests? Is it fair or can it cause unjust damage? Thirdly, does the model scale? May it be used in ever wider circumstances, at the same time reinforcing its hidden biases as it is applied in ever wider circumstances?

                      An example is the model used to determine sentence length in US courts based on a model about the risk of recidivism. The model takes into account factors, such as previous criminal record or age of first contact with the police, which would not be admissible as evidence in court and which may unfairly discriminate against certain sections of the population.

                      Chapter 2 talks about the role of algorithms in the financial crisis, noting two key issues that lead to the collapse in the mortgaged backed securities market. Firstly the assumption that models had been subject to proper mathematical vetting; in reality few people understood the mathematical and statistical issues and many that did lacked the integrity to speak up, especially as initial success had created its own feedback loop encouraging growth in the market. Secondly modern computing power had allowed a massive secondary infrastructure to grow around the market - credit default swaps, CDOs etc - that instead of diversifying the risk masked, magnified and concentrated it.

                      A very interesting point was made based on the author's experience in a risk assessment firm, that many traders are remunerated based on their Sharpe ratios, the ratio of revenues to risks taken, and accordingly are motivated to “...actively seek to underestimate…” risk in order to effectively manipulate the Sharpe ratio and hence their bonuses. This contrasts with the approach of hedge funds, which genuinely care about risks (given their own money is at risk) and traders at large financial institutions which don’t have their own financial capital at stake.

                      Chapter 3 looks at the impact of the algorithm developed by US News to rank colleges in the US. 75% of the ranking was based on proxy items intended to measure college success and 25% on subjective evaluation. The proxies selected, for example admission ratios, SAT scores, invited gaming that distorted applications; colleges would invest in sports in order to encourage applications that could be rejected; in an extreme case a new Saudi college required part time professors with large numbers of citations to change their site reference to the college in order to move up the rankings. The system does not measure the key success of education - what the students learnt at each school. The ranking generates its own feedback loop; colleges that rank high attract more applicants, generating more rejections thus moving them further up the rankings. Wealthy applicants pay consultants to game the system.

                      Crucially the original algorithm did not take into account college tuition costs. This guaranteed the early rankings being in line with existing “common sense” with Yale, Harvard and other wealthy colleges ranking high but thereby excluded an issue critical to students, value for money from an education, while encouraging colleges to spend excessively in order to improve student experience and hence ranking without regard to cost, ultimately leading to more student debt.

                      US govt has now made available data on schools allowing students to check directly.

                      Chapter 4 looks at targeted online advertising, taking for profit colleges that target poor and vulnerable people as a particularly nefarious case. The internet gives instant feedback for targeted marketing campaigns through Facebook or Google in particular where successful or failing ads can be evaluated in real time. Bayesian analysis is used to evaluate success.

                      20-30% of a for profit college’s budget may go on lead generation with more spent on recruitment than on education itself. Specialist lead generation firms exist targeting particular communities, posting fake job ads or promising health coverage.

                      Chapter 5 looks at “Predpol” an algorithm to predict crime which has a key input the geographic location of crime but which excludes data on race. The algorithm is successful in forecasting where crimes can occur, but mainly because it includes low level crimes - public drunkenness, jaywalking - which are better correlated with geographic location. However these crimes are also often associated with poverty and indirectly with race; more serious crimes are more difficult to detect while the system ignores some crimes altogether, such as white collar fraud. On the surface the algorithm is objective but under the hood it reflects value judgments around where the police direct their attention.

                      Issues of probable cause are also raised by the use of algorithms that predict whether a person will commit crime based on proxy data such as location of residence, whether or not employed or similar. These algorithms raise questions around to what extent the public is prepared to balance efficiency in police work against fairness, but without any public debate.

                      Chapter 6 looks at the use of algorithms in employee hiring decisions.. Personality tests are often used for screening job applicants, but these may be “run arounds” laws preventing discrimination against people with disabilities and, a key consideration for the correct use of algorithms, are rarely updated through monitoring their actual success in predicting good job performance. This may not be a problem for a system evaluating baseball stars, whose performance can be objectively statistically measured and where the individuals may be paid millions of dollars, rather the burden falls on lower paid staff in the collateral damage suffered by individuals that are screened out by the algorithms but are capable of doing the job.

                      These algorithms include negative feedback - those discriminated against fail to get good jobs, thus justifying the discrimination - and may be based on data that reflects historically discriminated hiring practices; an example is given of a system used in a British hospital that after many years of use was held to discriminate against women and immigrants, repeating discrimination reflected in the original data on which the system was based.

                      Chapter 7 examines the impact of algorithms in the workplace.

                      Job scheduling software uses data to schedule peaks and troughs of staffing, dehumanizing employees in the process who are unsure of their work schedule until only a day or two before being called on.

                      Another algorithm - ‘Cataphora’ - attempted to identify the most creative and innovative employees through tracking the flow of e-mail including certain key phrases through the e-mail system. There is little evidence that this approach worked, but employees who were not among those identified as the most creative may be first in line for termination. This algorithm suffers from the two classic problems identified early in the book, the difficulty of finding measurable proxy data for the items (soft skills such as creativity) that you want to measure and the lack of any feedback on success and failure of those measured to help the algorithm learn.

                      The chapter explained an egregious error in the evaluation of teacher performance through the use of SAT scores. The ‘Nation at Risk’ report issued by the Reagan administration was intended to address the decline in teaching standards as measured by a gradual decline in SAT scores of graduating students over the years. In fact the data illustrated an example of ‘Simpson’s Paradox’ in which the whole body of data showed one trend but when examined on a segmented trend the opposite trend was apparent. In this particular case the scope of people taking SAT tests had expanded over time including enough people at the lower end of the scoring range to lower the overall average (i.e. elite students had been sitting the test for many more years, so there was little room for scope to increase at the higher end of the scoring range). When the data was segmented in narrow SAT ranges SAT scores had increased in all segments over time. The whole premise of the report was false.

                      Chapter 8 looked at FICO scores. FICO scores themselves have some good features as data; they are based on an individual’s history (in contrast to being based on aggregate data inferred from people who resemble the individual), default on a loan is relatively objectively measured and feedback is used to improve scoring.

                      Problems arise when FICO scores are used as proxies for other data that is not so easily or objectively identified, for example future job success when used as screening in the hiring process, or when combined with other data in developing proprietary “e-numbers” used by firms for purposes such as marketing. These metrics are not transparent to consumers and risk being a backdoor for discrimination because they may use factors that are indirectly linked to poverty or race such as residence.

                      Chapter 9 considered how the whole insurance business model may be undermined by data algorithms. Insurance relies on the pooling of a wide range of risks, good and bad, to allow the pricing of the risk in aggregate. Algorithms allow such risks to be segmented in a non-transparent way which, in addition to undermining the insurance model itself by pricing out high risks, permits price gouging of other segments. The use of non risk related indicators, for example credit scores or location of residence, in pricing motor insurance also allows price gouging. Why should a drop in a driver’s credit rating impact his premium when the risk that he has an accident is unchanged? Drivers are being assessed on the consumer patterns of their friends and neighbors rather than on their accident record.

                      Chapter 10 looks at the application of algorithms to civic life, in particular the use of micro targeting of political messages through Facebook. Facebook has experimented with manipulating emotional responses of users through manipulating their newsfeeds.

                      Facebook’s approach to disseminating news is contrasted to a conventional newspaper in which the Editor makes a decision on what to put on the front page but that decision is seen by everyone and open to public debate whereas the criteria for selection of articles for a newsfeed by Facebook are opaque and unique to the viewer. Facebook micro targeting may be one of the reasons many Republican voters still believe Obama ‘birther’ and other conspiracies.

                      Micro targeting of voters risks disenfranchising everyone to the benefit of those paying for it. Voters in swing states are subject to more focused and intensive campaigns which are relevant only to themselves to the detriment of the democratic process for all.
                      current-affairs programming science
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                        Amin
                        396 reviews 405 followers
                  January 2, 2024
                      شاید بتوان عنوان کتاب را "سلاحهای کشتار جبری" ترجمه کرد تا منظور نویسنده بهتر حاصل شود، با این توضیح که منظور از جبر، معنای آن در ریاضی است و نه به معنای جبر و اختیار. فکر می کنم با محبوبیت این کتاب در دنیا و نقش رو به رشد داده پردازی در کشور ما دیر یا زود شاهد ترجمه آن باشیم.

                      بارها برای من پیش آمده که در فضای پیرامون که مملو از دانشجویان و محققین حوزه داده و مهندسی است بحثی پیش آید که مرزهای فنی و غیرفنی کار با داده چیست و تا چه حد پاسخ تمام مسائل به خودی خود از دل داده ها در می آید. یک بار کسی به من گفت دیگر نیازی به عامل انسانی نیست چون هوش مصنوعی خودش راه حل ها را می یابد و عرضه می کند. اما می دانیم که مثلا در کلان داده ها، به لحاظ آماری می شود بین پدیده های دلخواه بسیاری ارتباط معنادار یافت و جهت گیری محقق تعیین می کند که به چه ابعادی از آن معنا بدهد. به طور خلاصه اول مساله معنادار بودن نتایج را داریم و در مرحله بعد استفاده و تفسیر خاصی که از آنها می شود که در هر دو مرحله نقش عامل انسانی پررنگ است

                      این کتاب اما یک قدم هم فراتر می گذارد. الگوریتم هایی که نتایج تحلیل داده هایشان هم معنادار است و هم کاربردهای مشخصی دارد. اما گاهی خواسته و گاه ناخواسته معضلاتی درست می کنند که غالبا در راستای تخریب دموکراسی و افزایش فاصله طبقاتی هستند. با این که مثالها برای جامعه امریکا هستند، اما همین موقعیت جغرافیایی شاید بهترین مثال باشد تا وضعیتی رادیکال تر از استفاده از داده های بیشمار در تصمیم گیری های اجتماعی و سیاسی را ببینیم، البته با محوریت منافع اقتصادی و نقش شرکت ها. یک نکته مشترک هم در تمام این الگوریتم ها غیرشفاف بودن کارکرد آنها برای ناظران بیرونی و عدم وجود بازخورد برای اصلاح آنهاست که باعث شده نویسنده به هر کدام از آنها عنوان یک ابزار کشتار جبری بدهد

                      از سیستم رتبه بندی معلمان که گاه علیه معلمانی شایسته تر نتیجه می دهد و کل سیستم را بجای بهبود سیستم آموزشی به سوی افزایش امتیاز برای بقا در سییستم سوق میدهد، تا کارکرد شرکت های بیمه برای افزایش سودآوری که در نهایت به ضرر قشر آسیب پذیر عمل می کند، یا با همین منطق الگوریتم هایی که به پلیس کمک می کند تا خلافکاران بالقوه را شناسایی کند و در نتیجه زندان پر از افرادی می شود که غالبا از مناطق آسیب پذیر یا سیاه پوستان هستند. از این کاربردهای اجتماعی به سمت بازاریابی هدفمند که برویم پای دانشگاههای پولی و تبلیغات وسط می آید که به زبان نویسنده طبقه مرفه آگاه تر و باهوش تر از آن هستند که گول چنین سیستم هایی را بخورند؛ و در نهایت به سیستم اثرگذاری این الگوریتم ها در شبکه های اجتماعی میرسیم که اطلاعات دریافتی از سوی هر فرد را تحت تاثیر قرار می دهند تا جهت گیری سیاسی جمع زیادی از رای دهندگان مردد را قبل از زمان انتخابات تغییر دهند و به نوعی مانعی بر سر دموکراسی به شمار آیند

                      به نظرم فصل ها جذابیت های کم و زیاد متنوعی دارند. گاهی می توان با نظر نویسنده موافق نبود یا احساس کرد که در بیان نتیجه می تواند اغراق کند، اما در کل این کتاب را اثری خواندنی یافتم که با تخصص خود نویسنده که در همین باره است قابل اعتنا می شود
                      business politics science
                      25 likes
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                        Andy
                        1,682 reviews 547 followers
                  May 7, 2022
                      It's good to critique what the author calls WMDs, but for me this book missed the mark.

                      For example, the entire example of teacher-rating in DC is a red herring. The real question is: What is proven to deliver success in an urban US school system? (Answer: Hope and Despair in the American City: Why There Are No Bad Schools in Raleigh ) So it's valid to point out the silliness of a silly GIGO teacher-rating algorithm, but it's a distraction to fall down the vortex of dissecting that silliness in detail. We need to focus instead on facts about relevant outcomes. This is sorta kinda vaguely the theme the author seems to be developing but never fully formulates: the afterword delivers a pretty wishy-washy conclusion about the nature of objective truth.

                      Trying to say something nice... The flip side of the superficiality of this book is that it alights on numerous interesting factoids as it skims along. Also, it is clearly written.

                      Alternatives:
                      Moneyball: The Art of Winning an Unfair Game
                      The Big Short: Inside the Doomsday Machine
                      Detroit: An American Autopsy


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                        Chad Kohalyk
                        295 reviews 31 followers
                  October 3, 2016
                      Solid overview of the various mathematical models that govern our education, labour, wealth, and commerce. O'Neil packs in many examples and unpacks how simplistic, unfair and damaging to already disadvantaged people these models can be. As someone who worked on the front lines of developing models for predictive internet shopping, I was familiar with many of the tactics mentioned in this book, and their ethical shortcomings (which finally led to me leaving the business). What she says is entirely true, and it makes a small amount of people a lot of money. More people should be outraged. This book could help.

                      There are two shortcomings, though, both possibly due to its short length. First, there is no discussion at all of government abuses of all the data collection detailed within its pages. Maybe in the age of Snowden this is just assumed, but I think it is an important outcome of the big data revolution that should at least be addressed, even in passing. Secondly, I was hoping for more imaginative solutions. The "hippocratic oath" for data scientists she refers to is a nice idea... from twenty years and two big economic recessions ago. Advocating for stronger regulation is certainly prudent. However, I was hoping for something new.

                      Weapons of Math Destruction is a well documented tour of the standard examples of the misuse of math and big data, and concludes with the standard solutions. A good book to recommend to friends who need a primer on these issues that we have been facing for the last decade or so.
                      audio general-nonfiction
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                        Atila Iamarino
                        411 reviews 4,445 followers
                  February 7, 2017
                      Uma discussão muito boa e ponderada sobre o que acontece quando começamos a usar algoritmos para avaliar pessoas. A Cathy O'Neil passa por vários sistemas de avaliação automatizados e o tipo de viés que eles introduzem, quem fica de fora ou quem é mau avaliado. De sistemas de crédito a seguros, saúde, vagas de emprego e antecedentes criminais. Uma visão bastante importante do que está por vir com o dataísmo crescente (ver Homo Deus: A Brief History of Tomorrow ).

                      Não que isso seja um problema novo. Preconceito, premissas erradas e mais uma série de falhas desfavorecem muita gente desde sempre. Mas agora que podemos automatizar essa avaliação, o que eram vários humanos errando em várias direções vira um algoritmo errando consistentemente da mesma forma. O que pode aumentar o estrago feito, mas também dá margem para um aprimoramento muito mais eficiente.

                      A mensagem mais importante do livro me parece ser que, já que o uso dos nossos dados para tudo é inevitável, pelo menos saber o que está sendo levado em conta quando somos avaliados ajuda muito a previnir erros com consequências graves.
                      dataismo futuro tecnologia
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                        Jack Teng
                        Author 8 books 7 followers
                  October 7, 2016
                      I can't stress how important of a book this is. I don't think people really know how the obsession with Big Data and algorithms is about to control/influence our lives. I suppose I sound paranoid, but I really don't think I am. There are too many tinkerers out there who have some degree of competence at math and who think they'll solve all the world's problems with the next greatest optimization formula, and yet they lack even the most basic experience in asking proper research questions and understanding the major limitations of data. My own experience in theoretical ecology exposed me to empiricists wanting to get in on the modeling action and just produced garbage papers. Beware, people. Beware.
                      18 likes
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                        C.P. Cabaniss
                        Author 7 books 112 followers
                  February 20, 2017
                      *I received a copy of this book through Netgalley. All thoughts are my own.*

                      This book did not turn out to be what I was expecting. I expected O'Neil to go more in depth about the math behind data she was discussing, to explore the algorithms in greater detail. That was not what I got, however. This book turned out to be more of a superficial look at some of the ways big data can/has impacted society in the author's opinion. And while I found some of the material presented interesting and informative, I was overall not impressed by the arguments the author made. She didn't seem to have a clear idea for a resolution.

                      More thoughts will be up on my blog: www.courtneysreads.blogspot.com
                      nonfiction reviewed
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                        Murtaza
                        686 reviews 3,390 followers
                  January 21, 2018
                      A short and concise overview of the problems being wrought by the algorithms that are now quietly governing our society. While the overview is useful, I was a bit surprised at how little new information that there was in here. The issues that she raises should be familiar to anyone who has been following the impact of Big Data in even a cursory way. Facebook news bubbles, insurance profiles and other common phenomena are the main examples she uses. I did appreciate the social justice focus of the book, which makes sense given that the author is a former hedge fund analyst disenchanted by the malign behavior that she witnessed on Wall Street.

                      One key takeaway is that it there is a drive for poor and middle class people to have their lives processed by algorithms, while the rich will continue to depend on real-life personal connections to get by. In effect the masses are being turned into raw numbers with scores attached, while a small elite will continue to be authentic "personalities" who live in the more nuanced way.

                      This book lays out some of the ways to fight back against opaque algorithmic power and make these algorithms more fair. A solid overview but nothing groundbreaking.
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                        Margaret Sankey
                        Author 9 books 233 followers
                  September 16, 2016
                      I was really delighted to see this book on the list of nominations for the National Book Award--O'Neil was a math professor, then hedge fund analyst, then Occupy activist, now excellent non-fiction explicator of the power that we've given algorithms in daily life. Certainly, the old way, in which a human bank manager or college admissions officer selected people for loans or Harvard reflected significant prejudices--but have we gone too far in the other direction, thinking that hard data will erase biases? O'Neil offers vivid examples: microtargeted ads, teacher evaluations, police stats for predictive patrols, insurance, credit scores and the warping of higher ed around the US News and World Reports college rankings to demonstrate the potential for toxic grips of big data as opaque, unexplained, unchallenged and dictatorial, as well as the possibilities for using its power to help rather than exclude.
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                        Graeme Roberts
                        522 reviews 36 followers
                  November 3, 2021

                      Cathy O'Neil writes well and provides some good information, but I found the fundamental thesis of Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy to be foolish and irresponsible. The title, condensed inevitably to WMD, is used promiscuously to describe any application of data science, statistics, or even information technology that she considers to be unfair or discriminatory. In some cases, she may be right; some applications are used by the greedy and the uncaring with insufficient thought of the consequences, but she rarely considers the positive attributes of the applications. Once branded as a WMD, a term indelibly appended to the evil trifecta of George W. Bush, Saddam Hussein, and the Iraq War, how are readers meant to consider their many positive aspects and what can be done to make them better?

                      At first, I thought that Ms. O'Neil had been the victim of a foolish PR person in her publishing company who had come up with this catchy name, but her ardent use of the term convinced me that she had invented it.

                      Ms. O'Neil's own career seemed to follow a downward trajectory from her Harvard PhD in mathematics and a teaching position at Barnard to a leading hedge fund and a series of other ethically questionable environments that left her convinced that data science, and even mathematics, are roots of much evil. I don't buy it. They are tools that can be used or misused, and her diatribe does nothing to help.

                      She asserts, on page 210:

                      If we find (as studies have already shown) that the recidivism models codify prejudice and penalize the poor, then it's time to have a look at the inputs. In this case, they include loads of birds-of-a-feather connections. They predict an individual's behavior on the basis of the people he knows, his job, and his credit rating—details that would be inadmissible in court. The fairness fix is to throw out that data.

                      But wait, many would say. Are we going to sacrifice the accuracy of the model for fairness? Do we have to dumb down our algorithms?

                      In some cases, yes. If we're going to be equal before the law, or be treated equally as voters, we cannot stand for systems that drop us into different castes and treat us differently.

                      I disagree. Some data is better than none (provided it is accurate and not intentionally manipulative), though we should be looking to constantly improve our data science systems. I would like to know that the recently paroled criminal moving in next door had been released by a judge in full possession of whatever facts are available. So, I suspect, would Ms. O'Neil.
                      16 likes
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                        Lauma
                        224 reviews 1 follower
                  January 29, 2021
                      Datu zinātnieka obligātā literatūra.

                      Interesants ieskats dažādos matemātisko modeļu piemēros, kas cilvēku pieļautu kļūdu vai alkatības dēļ pasaulei dod vairāk slikta, nekā laba. Jebkurš modelis ir tik subjektīvs, cik subjektīvs ir tā veidotājs.

                      Mašīnām var iemācīt diskrimināciju.
                      Cilvēki jauc korelāciju ar cēloņsakarību.
                      Cilvēkiem patīk pelnīt uz citu cilvēku rēķina.

                      Autore norāda, ka bīstami matemātiksie modeļi ir mums apkārt un kontrolē mūsu dzīves daudz vairāk nekā mēs iedomājamies. Augstās mācību maksas ASV universitātēs? Pie vainas izglītības iestāžu tops, kurā samaksa par studijām nav iekļauta. Matemātiskie modeļi, kas paredz noziedzību pirms tā vēl notikusi? Ideja lieliska, izpildījums pieklibo, jo policistiem ir tieksme fokusēties uz maziem pārkāpumiem, kas bieži norisinās nabadzīgajos rajonos, kurus galvenokārt pārstāv ne tās gaišākās ādas krāsas pārstāvji. Jo vairāk tādiem pievērš uzmanību, jo vairāk modelis domā, ka tur notiek visaktīvākā noziedzība, kamēr bagāto un 'pieklājīgo' cilvēku rajonā vardarbību ģimenē gadiem neviens nepiefiksēs. Demokrātija? Cilvēkiem pielāgotas reklāmas for the win - dažādi cilvēki saņem dažādus solījumus no viena un tā paša politiķa. Facebook arī spēj ietekmēt cilvēku emocijas pietiekami, lai tas varētu izšķirt vēlēšanu dienu rezultātu. Vēlēšanu rezultāti nav jāvilto, pietiek ar pareizu cilvēku manipulāciju e-vidē, un matemātiskie modeļi to spēj nodrošināt.

                      Piemēru ir daudz, taču visus sliktos modeļus vieno 3 svarīgas īpašības - tie nav caurredzami, tie ietekmē lielas cilvēku grupas un to ietekme ir spēcīga ne tajā labākajā nozīmē. Nav tā, ka paši matemātiskie modeļi ir slikti, tie ir tikai instruments cilvēku rokās, un jebkurš instruments ir tik labs, cik labs un spējīgs ir tā izmantotājs. Veltot pietiekami daudz uzmanības sava darbam, jebkurš datu zinātnieks var parūpēties par to, lai viņa veidotais modelis nenonāktu šo piemēru sarakstgā un neradītu kaitējumu cilvēcei. Arī man ir gadījies upurēt modeļa precizitāti, no tā izņemot cilvēku radīto diskrimināciju datos. Jāatzīst, ka to izdarīt nemaz nav tik viegli, ja esi mācījies tiekties pēc iespējami precīzākā iznākuma, taču ir jāsaprot, ka ne vienmēr augstākais % skaitļos atbilst realitātei.

                      Sometimes the job of a data scientist is to know when you don't know enough.

                      Iesaku arī tiem, kas īpaši nedraudzējas ar matemātiku. Grāmata uzrakstīta pietiekami saprotamā valodā un vairāk iedziļinās tajā, kā strādā pasaule, kurā dzīvojam, pārāk neiedziļinoties matemātisko modeļu tehniskajās detaļās.
                      math-data-science nonfiction
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                        Nick Klagge
                        779 reviews 65 followers
                  February 5, 2017
                      Cathy O'Neil was my professor for number theory in college, I think in 2006. I thought she was a great teacher, but didn't keep in touch at all after the class. I was somewhat aware that she was involved with Occupy Wall Street's financial policy arm, and after I heard about this book, also learned that she had been co-hosting a podcast on Slate (which she is now about to leave!--but I still have a lot of back-episodes to listen to).

                      I'm broadly in agreement with the thrust of her argument in this book, and only have quibbles with regard to ways I think it necessarily simplifies things to be a popular-consumption nonfiction book. The main thrust of the book is that people frequently conflate "algorithmic decision-making" with "neutral decision-making", and that this is a fallacy (one that algorithm-purveyors are happy to perpetuate, for the most part). As "big data" and quantitative models get applied to more and more aspects of everyday life, it's incumbent on us to understand this, and to consider ways in which algorithmic decision-making can be problematic, biased, or dangerous.

                      O'Neil describes three features that characterize a "weapon of math destruction": scale, secrecy, and destructiveness. We mostly only need to focus on algorithms that have all three of these features, at least to some degree--a model that is in limited use, transparent, or harmless is not much of a cause for concern. She gives examples of algorithms in many fields that she sees as meeting all three of these criteria. One example is recidivism risk modeling, which is now used in many states to determine, at least in part, criminal sentencing. Errors or bias in these algorithms may result in additional years behind bars for the individuals they apply to, and they are both widespread and not publicly disclosed. There are many other interesting (/troubling) examples in the book, such as teacher value-added models.

                      An emergent property of many such algorithms is that they may engender undesirable feedback loops. For example, a recidivism risk model will be biased against black people if it is trained on historical data that cover an environment characterized by bias against black people. (If black people are more heavily prosecuted in general, they are likely to appear as higher recidivism risks, and even if the algorithm doesn't use race directly, it will pick up on correlated factors and amount to the same thing.) This is largely a function of the opacity characteristic--if an algorithm is publicly disclosed, people can bring scrutiny to it and highlight flaws.

                      The issue of model opacity is an especially interesting one to me, as someone who works on regulatory financial models that are intentionally not disclosed. There are strong reasons for not disclosing any models used for high-stakes decisions (in my case, setting minimum capital levels for banks). A primary concern is that a transparent algorithm will be "gamed", in the sense that those subject to it will figure out ways to make themselves "look good" to the algorithm that are driven more by the details and limitations of the model rather than by the underlying substance. A second concern is that a transparent regulatory model can encourage a "monoculture" in which those subject to it will simply adopt the model for themselves, rather than developing their own models that, while still flawed, will have different flaws than the regulatory model.

                      I don't think there is an obvious solution to this transparency problem. One solution that I definitely don't agree with is to eschew quantitative decision-making altogether. As O'Neil clearly states in the book, we shouldn't assume that pre-algorithmic decision-making was unbiased either--it seems quite apparent that, for example, there is bias in judgmental sentencing, perhaps more than in algorithmic sentencing. O'Neil herself has one proposed solution to this (which I don't think she really discusses in the book)--she has started a consulting company whose intent is to audit existing algorithms for potential biases or other damaging impacts. This would allow some degree of independent assessment while not disclosing the model generally. I think this is an interesting idea and I hope it takes off, but there are limitations. Especially in the private sector, it's not clear why a company would voluntarily request such an audit, especially if an algorithm is making them lots of money. Fear of regulatory penalties could be one motivation, but we're clearly entering an era of deregulation. Regulators themselves might force audits, but again that requires a strong regulator (also, who watches the watchers?).

                      One approach that I think could make sense in at least some contexts is a hybrid algorithmic-judgmental process. (At least right now, hybrid processes can be most effective in many fields--for example, the best chess player is neither a computer nor a human, but a human assisted by a computer.) To take the example of recidivism risk, we might have an algorithm that is publicly disclosed that produces a publicly-disclosed outcome to the judge (perhaps a recommended range). The judge may then choose to depart from the recommendation, but needs to give a written description of her reasoning for doing so. In this way, the algorithm can be audited by any outside party for potential biases, but the final judgmental step serves as insurance against flagrant cases of gaming the system, or cases with significant factors that are not considered by the algorithm.

                      As I said earlier, my only real quibbles with the book are around simplifications that I think are a reality of publishing a non-fiction book for popular consumption. For example, the terminology of "WMD" encourages us to think in a binary way (is it or isn't it one?), rather than seeing a continuum from OK to troubling, which I think is a better reflection of reality.

                      Finally, I'll add that this book, published in September 2016, proved to be quite prescient. The public controversy about "fake news" and the Facebook newsfeed algorithm arose shortly after its publication. Interestingly, I think before this happened, few would have identified the newsfeed algorithm as a potential WMD, because the vector for "destructiveness" was non-obvious. O'Neil has written some articles on this topic since the publication of the book, which are worth looking up.
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                        Jill
                        2,203 reviews 93 followers
                  September 4, 2016
                      This book shows the hidden ways in which the use of "Big Data" is much more far-reaching and harmful than expected. Big data refers to the massive amount of information now available because of computers that is collected and analyzed and sold to third parties.

                      In particular, as the author demonstrates convincingly, applications of Big Data “punish the poor and the oppressed in our society, while making the rich richer.” She paints a sobering picture.

                      The author calls the mathematical models employing Big Data and used to such harmful effect “Weapons of Math Destruction” or WMDs.

                      In WMDs, she explains, “poisonous assumptions . . . camouflaged by math go largely untested and unquestioned.” They create their own toxic feedback loops, and, to an extent which shocked me, guide decisions in a large variety of areas ranging from advertising to prisons to healthcare to hiring and firing decisions. Most importantly, because they rely on esoteric mathematical models, no matter that they are many times based on biased and/or erroneous premises:

                      “They’re opaque, unquestioned, and unaccountable, and they operate at a scale to sort, target, or ‘optimize’ millions of people.”

                      The goal is always profit, but what is lost is fairness, the recognition of individual exceptions, and simple compassion and humanity, adding to the inequality gap, not to mention downward spirals for some unfortunate victims from which it is almost impossible to escape.

                      I am not at all well-versed in math, but the author manages to explain how all this works without requiring that one understand specific algorithms. She provides examples from the worlds of teacher evaluations, hiring decisions generally, advertising, insurance, police programs, college admissions, lending and credit evaluation, and political targeting.

                      One of the saddest chapters (and they are all sad, unfortunately) is about the many for-profit universities (Trump University comes to mind) that specifically target people in great need, selling them overpriced promises of success. Her quotes from the marketing materials of these places are horrifying. They look for individuals who are “isolated,” with “low self esteem” who have “few people in their lives who care about them” and who feel “stuck.” She shows how they use google searches, residential data, and Facebook posts, inter alia, to find “the most desperate among us at enormous scale”:

                      “In education, they promise what’s usually a false road to prosperity, while also calculating how to maximize the dollars they draw from each prospect. Their operations cause immense and nefarious feedback loops and leave their customers buried under mountains of debt.”

                      The chapter on the way the “stop and frisk” policing operates is also very depressing; and in truth we have seen the tragic results in city after city.

                      The fact is, the whole book is rather a downer, albeit an important one. Although O’Neil cites a few programs that have used Big Data to help people rather than to enrich a few and oppress the rest, can one really think that “moral imagination” can take precedence over prejudice and greed? Personally, I’m not so sure. The author provides ideas about how to change (and importantly, regulate) uses of Big Data, but she is more optimistic than I am, ending on a positive note:

                      “We must come together to police these WMDs, to tame and disarm them. My hope is that they’ll be remembered, like the deadly coal mines of a century ago, as relics of the early days of this new revolution, before we learned how to bring fairness and accountability to the age of data. Math deserves much better than WMDs, and democracy does too.”

                      Evaluation: I hope this important book gets a lot of attention. My husband always makes the argument about privacy concerns that what do we care if we’ve done nothing wrong? This book shows how, astoundingly, that isn’t enough to stop Big Data from hurting us in many aspects of our lives. It is a critical lesson for today’s world, and the world of our children.
                      14 likes
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                        Kurt Pankau
                        Author 11 books 21 followers
                  November 1, 2016
                      My father, in one of his grouchier get-off-my-lawn moments, complained to me about computers in the workplace. Specifically, it bothered him that people were so trusting in the software that they were unwilling to gut-check outputs. This conversation happened maybe twenty years ago, but it stuck with me, and part of the reason is because he was absolutely right. We've all heard the phrase "garbage in, garbage out" to describe this phenomenon.

                      O'Neil takes the idea of "garbage in, garbage out" and compounds it, and this book is an exploration of the pitfalls, both real and potential, involved in letting algorithms dictate our lives, in everything from teacher retention to prison recidivism to workplace wellness programs. I'm a huge fan of O'Neil's contributions to the Slate Money podcast, which is how I learned of this book in the first place. And if you're at all interested in the Big Data economy, you should go read it right now. Even if you're afraid of math--in fact, especially if you're afraid of math. O'Neil's writing is not overly technical. This book is a very easy read and it is dense with ideas.

                      Many of these ideas resonated the instant they hit my brain, making me recognize things that had been around me for months or years. To give an example, O'Neil has a chapter on insurance and describes how more insurers are using e-scores based on "birds-of-a-feather" logic and demographic information. She points out the extent to which these things can become ridiculous, where having a drunk driving conviction can result in LOWER car insurance premiums than having a bad credit rating. While reading this, I remembered the "Sorta Marge" commercial Esurance put out and thought "Oh, that's what that was about."

                      There's a big social justice component to this book that hinges on the fact that algorithms sacrifice individuals in favor of trends, and in doing so they create feedback effects. Another example: some states include the defendant's zip code in sentencing models. This would be inadmissible in a court case, because it's a proxy for race and/or income and is irrelevant to the individuals guilt or innocence. But sentencing modelers argue that it accurately predicts how likely they are to be a repeat offender. So a person from the "wrong" zip code gets a longer sentence, then it's harder for him/her to find a place in society after release, and they end up going back to jail. The model doesn't just predict, it reinforces, and this is a problem. Because inherent in these models is an idea that O'Neil circles back on again and again: since we use these models to help shape the future, they have to reflect the kind of future we want to live in rather than just codifying the past.

                      It's another book I desperately want everyone I know to go out and read so we can have lengthy discussions about it in bars. Here's my only real gripe: I hate the phrase "Weapons of Math Destruction." It's an excellent title and a key concept in the book, but O'Neil uses "WMD" as shorthand throughout. Every time I saw it, I had to remind myself that it meant "Math" destruction instead of "Mass" destruction, and by about Chapter 5 the joke had stopped being funny to me. End rant.

                      I loved this book and I will no doubt be reading it again. I could talk about it for a long time, so I'm just going to stop. But, as O'Neil points out, we're in the early days of Big Data. These mathematical models are affecting your life whether you realize it or not, and they don't have to be pernicious. But now is the time to start informing yourself if you want to help shape the conversation tomorrow.
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                        Wick Welker
                        Author 7 books 520 followers
                  December 9, 2020
                      Big data codifies prejudice and penalizes the poor.

                      A decent take on the "garbage in-garbage out" theory of data utilization. Cathy O'Neil appears aptly qualified to comment on the phenomenon of data models becoming an amplified version of societies biggest problems. O'Neil has invented her own term, Weapons of Math Destruction (WMDs) to lift the veil on a hidden but very active phenomenon that is perpetrating the racial and class biases that are already prevalent.

                      Per O'Neil, WMDs are any data algorithm that is used by any governmental or private organization to increase output, decrease waste and offload the critical thinking from humans to computers. The problem with WMDs is that they are not dynamic models that adapt and change based on a feedback loop. Instead, old data is feed into these machines which then crunch numbers and output a result that is considered sacrosanct by whoever is using the algorithm. The WMD almost becomes like a mystic--you cannot question the WMD, it is beyond repudiation. Yet the output is opaque and the result compounds existing problems.

                      Examples abound. School districts use WMDs to determine who the "bad teachers" are to then get rid of them. Yet these programs will score one teacher at the bottom one year and at the top the next year in an arbitrary way. The issue is that a teacher has only a few number of students, not say 10,000. These low numbers make the data amplify noise, not signal. Other models like baseball statistics can be effective because it is wholly transparent, where WMDs are not, using proprietary algorithms. Bad models are opaque, blind to pertinent information and are not dynamic.

                      Data models are a black box beyond scrutiny and have a positive feedback loop within themselves that serves as self prophecy. The US world and News report offers a great example of becoming the standard upon which college data models have been fit. The metrics become the target , completely invalidating the list to begin with. The cost of education isn't put into the formula which creates a "gilded checkbook" for colleges. Tuition has sky-rocketed as a result of this phenomenon. Predatory for-profit colleges take advantage of the vulnerable by using WMDs on social media.

                      Geography used in crime WMDs are a surrogate for poverty and race because cities are highly racially segregated to begin with. Data models for lending, insurance and employment based on zip code amplify racial wealth disparities. Prison recidivism rates based on WMDs have the exact same problem. Scheduling practices for workers using WMDs create unstable work life balances for the impoverished. Credit scores used as proxies for hiring compound the positive feedback loop of poverty and debt. Credit is a marker of wealth and wealth is a marker of race. All of these data go into WMDs and amplify the problems once again. There is asymmetry in political micro targeting thanks to WMDs.

                      Overall, informative, if not a tad outdated (written 2016) read about the misuse of bad data. I recommend.
                      nonfiction
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                        Tanja Berg
                        2,051 reviews 491 followers
                  February 25, 2018
                      Much of our lives are now influenced by algorithms. These are deemed to be neutral, but in fact many are based on models that are glaringly discriminatory. Zip codes stand in for race, for example. The poor are being particularly targeted for pay-day loans. Teacher are measured on opaque proxies that do no reflect their skills at all. Your credit score is as much a result of the group of people you are thrown in with, as with your own behavior. These algorithms are what the author calls "weapons of math destruction".

                      Every on-line decision you take, every click you make, is being monitoried. Your ads are tailored, and so is your Facebook news feed. "Big brother sees you" has a whole new meaning since "1984" was written 50 years ago. We willingly leave a trail of information across a range of sites. We take stupid tests on FB that sells information on ourselves and our friends to the highest bidder. Some times it seems that you only need to consider making an investment for the ads to start trailing your feed and your online existence.

                      Turns out that in making hiring decisions, intelligence tests are prohibited - that is why personality tests flourish. Personality is being used as a proxy for intelligence. This despite the fact that there is little correlation between personality and job performance and that only one trait - conscientiousness - bears any correlation to work performance at all. Job seekers aren't told when their personality test red-flags them even for minimum wage jobs.

                      This book reinforces my notion that we're all screwed. Still, awareness that the internet bubble we all live in does not represent the ultimate truth, would be beneficial.

                      non-fiction science
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                        Aaron
                        203 reviews 43 followers
                  March 3, 2017
                      We've all seen the Big Data books: the future is now! A/B testing forever! AlphaGo crushed it! OkCupid says you shouldn't have a shirtless fish pic, you adorably dull redneck!

                      But Big Data has a darkside, and O'Neil goes through each segment of our life to show how these "models" can be used against us, to extract goods from us, and to keep us poor. Unfortunately, she also loses her argumentative power that could come with nuance, and she has to disregard nuance in order to make it understandable to the layperson (in other words, I don't think she's very charitable to the layperson).

                      The first big issue she brings up and has lots of evidence for is: Transparency. Lots of Big Data models that we use (and are fed into) on a daily basis, are not transparent. They're opaque equations sitting on a server farm. A teacher being graded on a value added model doesn't know where their score is coming from. A potential hire doesn't know why he failed his psyche evaluation. A criminal standing before judge doesn't know why the score said he'd be more like to return to jail.

                      This is honestly one of my biggest takeaways, and it hits close to home. When you apply for a credit card, they tell you why you failed to get one. You can access your "data point" and know why you have the FICO score you do. Google and Facebook tell you the metrics they use to serve you ads in their settings. I know that Amazon is trying to get me to buy another wallet even though I just bought a wallet because they showed me five hundred ads for another wallet. But what about those ads that Forbes tries to serve me? The cookies sitting on my computer watching me? I have no ideas what they're doing and not readily known way to find out. What about non-regulated credit systems that exist out there in Web 2.0 land? Bank of America controls for race and tries to stop redlining when they make a new policy, but will Peter Thiel try to do that when he invests in E-Corp? Probably not.

                      The second big problem with some Big Data systems is that they create feedback loops that increase inequality. Here, O'Neil is super weak except when she brings up the criminal justice example- we could be using big data to help keep people out of prison and make programs that lower recidivism, but instead we're using it as a way to keep white people out of prison.... but am I really supposed to believe that ads help make poor people poorer? She brings up for-profit schools using targeted ads to lure immigrants and poor people into massive student debt to make a profit, and, while I admit that's super shady, it's not the targeted ads' fault, is it?

                      The third problem with these "Weapons of Math destruction" is that they often have skewed data. This is the old line "garbage in, garbage out" except now it's "racist/sexist garbage in, racist/sexist garbage out." For example, if you make a employment system that filters out resumes, than teach it on a bunch of older resumes, you're inputting the bias of those older resumes. So if the guy that was reading those resumes was racist, you might be teaching a racist model.

                      Technically O'Neil has two other "bad points" about "WMDs" but they're just about scale.

                      Now, there are a lot of problems in this book and O'Neil kind of goes on tangents. For one thing, the WMDs she brings up are less "weapons" than they are symptoms of a bigger societal problem. Take "democracy": our current political system allows a few people- those that live in Orlando, Florida and Pennsylvania, basically- to chose who will be the President. This is messed up, but it means that the Democratic Party could build a powerful machine learning system that most efficiently spent money in locations to help change hearts and minds and win. She really dislikes this Big Data system, and says it's a threat to democracy...

                      ... but the electoral system itself is giant problem and threat to democracy (see: election of 2000)! Big Data has nothing to do with it!

                      The book ought to have been longer, and it ought to have included more counterexamples of positive data models (I can recall only two, FICO and some housing model). I think that she should've, if not had hand written equations or step-by-step instructions, at least given some background on actual data science. The way it is written makes it seem like she's a magician-mathematician that wandered down from the Ivory tower and realized that bankers were using magic for evil and now she wants to raise hell.

                      But I guess if I wanted authors to stop writing popular non-fiction books that they A/B tested on their blogs and turned into TED talks, I should stop reading popular non-fiction.
                      non-fiction statistics
                      12 likes
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                        Krysten
                        492 reviews 20 followers
                  September 29, 2016
                      "Big Data processes codify the past. They do not invent the future. Doing that requires moral imagination, and that's something only humans can provide."

                      So, for work, I read a lot about big data. A lot. And it's all basically *jackoff motion* uhhnnnggg big dataaaa unnnnnnngggghhh yeah. And that makes me want to die.

                      This book is refreshingly critical of big data and algorithms, from a blessedly human approach. You might expect a lot of statistics and dryness, but it's a lot of real life stories to illustrate O'Neil's arguments, which makes it easy to read and care about. It's also a fairly quick read and doesn't waste a lot of time defining concept. O'Neil gets right to it, and her arguments are powerful. If I ever wonder why I can't seem to get my credit card APR down to a reasonable level even though my credit score is excellent, I have to think - maybe it's my zip code! So many pieces of data are taken into account for so many things that end up affecting poor people and people of color in wildly disproportionate ways. You might pay 5x more for auto insurance because you drive through "bad" zip codes at certain times of the day.

                      This book is pretty amazing and I'm so, so glad I read it.
                      nonfiction
                      13 likes
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Weapons of Math Destruction: How Big Data Increases Ine…
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              A Survey of Mathematics with Applications (10th Edition) - Standalone book 10th Edition

              by Allen Angel (Author), Christine Abbott (Author), Dennis Runde (Author)
                  4.5 4.5 out of 5 stars 411 ratings
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                Note: You are purchasing a standalone product; MyLab & Mastering does not come packaged with this content. Students, if interested in purchasing this title with MyLab & Mastering, ask your instructor for the correct package ISBN and Course ID. Instructors, contact your Pearson representative for more information. If you would like to purchase boththe physical text and MyLab & Mastering, search for: 0134115767 / 9780134115764 * A Survey of Mathematics with Applications plus MyMathLab Student Access Card -- Access Code Card Package Package consists of: 0134112105 / 9780134112107 * A Survey of Mathematics with Applications 0321431308 / 9780321431301 * MyMathLab -- Glue-in Access Card 0321654064 / 9780321654069 * MyMathLab Inside Star Sticker
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                "For courses covering general topics in math course, often called liberal arts math, contemporary math, or survey of math." Everyday math, everyday language. The Tenth Edition of "A Survey of Mathematics with Applications " continues the tradition of showing students how we use mathematics in our daily lives and why it s important, in a clear and accessible way. With straightforward language, detailed examples, and interesting applications, the authors ensure non-majors will relate to the math and understand the mathematical concepts that pervade their lives. With this revision, an expanded media program in MyMathLab, and a new workbook further build upon the tradition of motivating and supporting student learning. Also available with MyMathLab MyMathLab is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. Within its structured environment, students practice what they learn, test their understanding, and engage with media resources to help them absorb course material and understand difficult concepts. NEW! This edition s MyMathLab course provides additional tools to help with understanding and preparedness.
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                             1. ISBN-10
                                0134112105
                             2. ISBN-13
                                978-0134112107
                             3. Edition
                                10th
                             4. Publisher
                                Pearson
                             5. Publication date
                                January 3, 2016
                             6. Language
                                English
                             7. Dimensions
                                8.6 x 1.5 x 11 inches
                             8. Print length
                                1024 pages
                           9. See all details
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          Editorial Reviews

            About the Author

              Allen Angel received his BS and MS in mathematics from SUNY at New Paltz. He completed additional graduate work at Rutgers University. He taught at Sullivan County Community College and Monroe Community College, where he served as chairperson of the Mathematics Department. He served as Assistant Director of the National Science Foundation at Rutgers University for the summers of 1967 - 1970. He was President of The New York State Mathematics Association of Two Year Colleges (NYSMATYC). He also served as Northeast Vice President of the American Mathematics Association of Two Year Colleges (AMATYC). Allen lives in Palm Harbor, Florida but spends his summers in Penfield, New York. He enjoys playing tennis and watching sports. He also enjoys traveling with his wife Kathy.

              Christine Abbott received her undergraduate degree in mathematics from SUNY Brockport and her graduate degree in mathematics education from Syracuse University. Since then she has taught mathematics at Monroe Community College and has recently chaired the department. In her spare time she enjoys watching sporting events, particularly baseball, college basketball, college football, and the NFL. She also enjoys spending time with her family, traveling, and reading

              Dennis Runde has a BS degree and an MS degree in Mathematics from the University of Wisconsin―Platteville and Milwaukee, respectively. He has a PhD in Mathematics Education from the University of South Florida. He has been teaching for more than twenty-five years at State College of Florida―Manatee-Sarasota. His interests include reading, history, fishing, canoeing, and cooking. He and his wife Kristin stay busy keeping up with their three sons–Alex, Nick, and Max.

            Product details

                * Publisher ‏ : ‎ Pearson; 10th edition (January 3, 2016)
                * Language ‏ : ‎ English
                * Hardcover ‏ : ‎ 1024 pages
                * ISBN-10 ‏ : ‎ 0134112105
                * ISBN-13 ‏ : ‎ 978-0134112107
                * Item Weight ‏ : ‎ 4.72 pounds
                * Dimensions ‏ : ‎ 8.6 x 1.5 x 11 inches
              * Best Sellers Rank: #120,173 in Books ( See Top 100 in Books )
                  + #266 in Mathematics (Books)
              * Customer Reviews:
                  4.5 4.5 out of 5 stars 411 ratings
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Amazon.com: A Survey of Mathematics with Applications (10th Edition) - Standalone book: 0000134112105: Angel, Allen, Abbott, Christine, Runde, Dennis: Books
https://www.amazon.com/Survey-Mathematics-Applications-10th-Standalone/dp/0134112105
None
2023-03-07T00:00:00Z
<div><div><p>1. Critical Thinking Skills<br/>
1.1 Inductive and Deductive Reasoning<br/>
1.2 Estimation Techniques<br/>
1.3 Problem-Solving Procedures </p><p>

2. Sets<br/>
2.1 Set Concepts<br/>
2.2 Subsets<br/>
2.3 Venn Diagrams and Set Operations<br/>
2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets<br/>
2.5 Applications of Sets<br/>
2.6 Infinite Sets </p><p>

3. Logic<br/>
3.1 Statements and Logical Connectives<br/>
3.2 Truth Tables for Negation, Conjunction, and Disjunction<br/>
3.3 Truth Tables for the Conditional and Biconditional<br/>
3.4 Equivalent Statements<br/>
3.5 Symbolic Arguments<br/>
3.6 Euler Diagrams and Syllogistic Arguments<br/>
3.7 Switching Circuits </p><p>

4. Systems of Numeration<br/>
4.1 Additive, Multiplicative, and Ciphered Systems of Numeration<br/>
4.2 Place-Value or Positional-Value Numeration Systems<br/>
4.3 Other Bases<br/>
4.4 Perform Computations in Other Bases<br/>
4.5 Early Computational Methods </p><p>

5. Number Theory and the Real Number System<br/>
5.1 Number Theory<br/>
5.2 The Integers<br/>
5.3 The Rational Numbers<br/>
5.4 The Irrational Numbers<br/>
5.5 Real Numbers and Their Properties<br/>
5.6 Rules of Exponents and Scientific Notation<br/>
5.7 Arithmetic and Geometric Sequences<br/>
5.8 Fibonacci Sequence </p><p>

6. Algebra, Graphs, and Functions<br/>
6.1 Order of Operations and Solving Linear Equations<br/>
6.2 Formulas<br/>
6.3 Applications of Algebra<br/>
6.4 Variation<br/>
6.5 Solving Linear Inequalities<br/>
6.6 Graphing Linear Equations<br/>
6.7 Solving Systems of Linear Equations<br/>
6.8 Linear Inequalities in Two Variables and Systems of Linear Inequalities<br/>
6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula<br/>
6.10 Functions and Their Graphs </p><p>

 
7. The Metric System<br/>
7.1 Basic Terms and Conversions Within the Metric System<br/>
7.2 Length, Area, and Volume<br/>
7.3 Mass and Temperature<br/>
7.4 Dimensional Analysis and Conversions to and from the Metric System </p><p>

8. Geometry<br/>
8.1 Points, Lines, Planes, and Angles<br/>
8.2 Polygons<br/>
8.3 Perimeter and Area<br/>
8.4 Volume and Surface Area<br/>
8.5 Transformational Geometry, Symmetry, and Tessellations<br/>
8.6 Topology<br/>
8.7 Non-Euclidean Geometry and Fractal Geometry </p><p>

9. Mathematical Systems<br/>
9.1 Groups<br/>
9.2 Finite Mathematical Systems<br/>
9.3 Modular Arithmetic
9.4 Matrices </p><p>

10. Consumer Mathematics<br/>
10.1 Percent<br/>
10.2 Personal Loans and Simple Interest<br/>
10.3 Compound Interest<br/>
10.4 Installment Buying<br/>
10.5 Buying a House with a Mortgage<br/>
10.6 Ordinary Annuities, Sinking Funds, and Retirement Investments </p><p>

11. Probability<br/>
11.1 Empirical and Theoretical Probabilities<br/>
11.2 Odds<br/>
11.3 Expected Value (Expectation)<br/>
11.4 Tree Diagrams 678<br/>
11.5 OR and AND Problems<br/>
11.6 Conditional Probability<br/>
11.7 The Fundamental Counting Principle and Permutations<br/>
11.8 Combinations<br/>
11.9 Solving Probability Problems by Using Combinations<br/>
11.10 Binomial Probability Formula </p><p>

12. Statistics<br/>
12.1 Sampling Techniques and Misuses of Statistics<br/>
12.2 Frequency Distributions and Statistical Graphs<br/>
12.3 Measures of Central Tendency and Position<br/>
12.4 Measures of Dispersion<br/>
12.5 The Normal Curve<br/>
12.6 Linear Correlation and Regression </p><p>

13. Graph Theory<br/>
13.1 Graphs, Paths, and Circuits<br/>
13.2 Euler Paths and Euler Circuits<br/>
13.3 Hamilton Paths and Hamilton Circuits<br/>
13.4 Trees 864 14 Voting and Apportionment <br/>
14. Voting and Apportionment<br/>
14.1 Voting Methods<br/>
14.2 Flaws of the Voting Methods<br/>
14.3 Apportionment Methods<br/>
14.4 Flaws of the Apportionment Methods</p></div></div>||||I||||                          * Product Catalogs
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                     1. Savvas Learning Company
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                     4. A Survey of Mathematics with Applications
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                      A Survey of Mathematics with Applications, 11th Edition © 2021

                        A Survey of Mathematics with Applications, published by Pearson, uses straightforward language and clear examples to reach students in the liberal arts, social sciences, business, and health fields and shows them how math ties into their interests and everyday lives.

                          * New animations let students interact tangibly with math
                          * New downloadable data sets for problems and examples
                          * Empower students through personalized learning
                          * MyMathLab® for School from Pearson flexible teaching and learning platform
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                        High School Math Curriculum with Digital Resources

                          Make math personal for students focused on other disciplines with a math program that adapts to their needs.

                            Relevant Learning

                            The authors have added Why This Is Important Notes when presenting concepts to help students connect their lives to the math they are learning.

                            Problem Solving and Exam Support

                            Beginning in Chapter 1, students are introduced to problem solving and critical thinking. Special problem-solving exercises in the exercise sets weave this theme throughout the entire textbook.

                            Learn by Doing and Get Up to Speed

                            The list of prerequisite skills needed for each section are included in the Annotated Instructor Edition and the MyMathLab for School course along with assignable assessments and remediation.

                            Adjustable Pedagogy

                            The program’s online hub provides easy access to courses and assignments, usage stats and goals, announcements, and helpful tips on using the tools.

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                        A Survey of Mathematics Teaching Solutions

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                                      * Engaging and Relevant Material
                                      * Developing a Thoughtful Approach
                                      * Opportunities to Practice Prerequisite Skills
                                      * Online Platform Features

                                            Engaging and Relevant Material

                                                                  * Data and Contexts
                                                                  Data and their contexts of application have been updated for current relevance.
                                                                  * Why This Is Important Section
                                                                  Interesting and motivational applications introduce each chapter, including the Why This Is Important section, illustrating the real-world nature of the topic.
                                                                  * Did You Know? Features
                                                                  Colorful and engaging Did You Know? Features highlight the connections of mathematics to history, the arts and sciences, technology, and other disciplines.
                                                                  * Recreational Math Features
                                                                  Recreational Math features invite students to apply the math in puzzles, games, and brain teasers. These features can also be assigned as homework.
                                                                  * Profiles in Mathematics
                                                                  Brief Profiles in Mathematics sketches and vignettes profile a diverse selection of historical figures who have advanced the discipline of mathematics.

                                            Developing a Thoughtful Approach

                                                                  * Crucial Thinking Skills
                                                                  Sections on inductive and deductive reasoning, estimation, and dimensional analysis build crucial thinking skills as students work through the book.
                                                                  * Key-Idea Boxes
                                                                  Key-Idea Boxes call out important definitions, formulas, and procedures in ways that are easy for students to identify.
                                                                  * Timely Tips
                                                                  Timely Tips boxes offer helpful information to make the material under discussion more understandable.
                                                                  * Summary Charts
                                                                  End-of-chapter summary charts provide an easy study experience by directing students to the location in the text where specific concepts are discussed. Review Exercises and Chapter Tests help students go over material again and prepare for exams.

                                            Opportunities to Practice Prerequisite Skills

                                                                  * New Data Sets
                                                                  New data sets can be downloaded in StatCrunch, .txt, and .csv file formats for problems and examples requiring analysis and marked with a Data Set icon.
                                                                  * Conceptual Enrichment
                                                                  Animations and StatCrunch icons throughout the narrative of the text encourage students to get hands-on with the topic and enrich their understanding of concepts. These new activities can be used in class or for independent exploration.
                                                                  * Warm-Up Exercises
                                                                  Exercise sets begin with fill-in-the-blank Warm-Up Exercises. Other exercises include Practice the Skills, Problem Solving, Challenge Problems/Group Activities, Recreational Mathematics, and Research Activities.
                                                                  * Technology Tips
                                                                  Revised Technology Tips go beyond graphing calculators and spreadsheets. Content now references apps on smartphones and tablets to explore various mathematical concepts and solve application problems.

                                            Online Platform Features

                                                                  * Integrated Review
                                                                  The updated Integrated Review provides embedded and personalized review of prerequisite topics within relevant chapters.
                                                                  * Personal Inventory Assessments
                                                                  New Personal Inventory Assessments promote self-reflection and engagement in students. The 33 assessments include topics such as Stress Management, Diagnosing Poor Performance, Enhancing Motivation, and Time Management.
                                                                  * StatCrunch Activities
                                                                  StatCrunch activities facilitate active learning and exploration through a powerful web-based statistical software that can be used to analyze and understand statistical concepts.
                                                                  * Comprehensive Video Program
                                                                  A comprehensive video program walks students through the concepts from every section of the text in a modern presentation format. Videos can be assigned and understanding tested via the accompanying video assessment questions.
                                                                  * Mindset Module
                                                                  A new mindset module is available in the course with growth-mindset-focused videos and exercises. Students develop a positive attitude about learning, value their own ability to grow, and view mistakes as a learning opportunity.

                        Empower Your Math Students with the MyMathLab® Platform from Pearson

                            Request a Demo

                            MyMathLab® for School from Pearson empowers students and teachers with adaptive resources that can be used in a blended or 100% digital classroom.

                          More About A Survey of Mathematics with Applications

                                                * Allen Angel Author Bio
                                                Allen Angel received his BS and MS in mathematics from SUNY at New Paltz. He completed additional graduate work at Rutgers University. He taught at Sullivan County Community College and Monroe Community College, where he served as chairperson of the Mathematics Department. He served as Assistant Director of the National Science Foundation at Rutgers University for the summers of 1967 - 1970. He was President of The New York State Mathematics Association of Two Year Colleges (NYSMATYC). He also served as Northeast Vice President of the American Mathematics Association of Two Year Colleges (AMATYC). Allen lives in Palm Harbor, Florida but spends his summers in Penfield, New York. He enjoys playing tennis and watching sports. He also enjoys traveling with his wife Kathy.
                                                * Christine Abbott Author Bio
                                                Christine Abbott received her undergraduate degree in mathematics from SUNY Brockport and her graduate degree in mathematics education from Syracuse University. Since then she has taught mathematics at Monroe Community College and has recently chaired the department. In her spare time she enjoys watching sporting events, particularly baseball, college basketball, college football, and the NFL. She also enjoys spending time with her family, traveling, and reading.
                                                * Dennis Runde Author Bio
                                                Dennis Runde has a BS degree and an MS degree in Mathematics from the University of Wisconsin–Platteville and Milwaukee, respectively. He has a PhD in Mathematics Education from the University of South Florida. He has been teaching for more than twenty-five years at State College of Florida–Manatee-Sarasota. His interests include reading, history, fishing, canoeing, and cooking. He and his wife Kristin stay busy keeping up with their three sons—Alex, Nick, and Max.
                                                * Table of Contents
                                                1. Critical Thinking Skills
                                                1.1 Inductive and Deductive Reasoning
                                                1.2 Estimation Techniques
                                                1.3 Problem-Solving Procedures

                                                2. Sets
                                                2.1 Set Concepts
                                                2.2 Subsets
                                                2.3 Venn Diagrams and Set Operations
                                                2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets
                                                2.5 Applications of Sets
                                                2.6 Infinite Sets

                                                3. Logic
                                                3.1 Statements and Logical Connectives
                                                3.2 Truth Tables for Negation, Conjunction, and Disjunction
                                                3.3 Truth Tables for the Conditional and Biconditional
                                                3.4 Equivalent Statements
                                                3.5 Symbolic Arguments
                                                3.6 Euler Diagrams and Syllogistic Arguments
                                                3.7 Switching Circuits

                                                4. Systems of Numeration
                                                4.1 Additive, Multiplicative, and Ciphered Systems of Numeration
                                                4.2 Place-Value or Positional-Value Numeration Systems
                                                4.3 Other Bases
                                                4.4 Perform Computations in Other Bases
                                                4.5 Early Computational Methods

                                                5. Number Theory and the Real Number System
                                                5.1 Number Theory
                                                5.2 The Integers
                                                5.3 The Rational Numbers
                                                5.4 The Irrational Numbers
                                                5.5 Real Numbers and Their Properties
                                                5.6 Rules of Exponents and Scientific Notation
                                                5.7 Arithmetic and Geometric Sequences
                                                5.8 Fibonacci Sequence

                                                6. Algebra, Graphs, and Functions
                                                6.1 Order of Operations and Solving Linear Equations
                                                6.2 Formulas
                                                6.3 Applications of Algebra
                                                6.4 Variation
                                                6.5 Solving Linear Inequalities
                                                6.6 Graphing Linear Equations
                                                6.7 Solving Systems of Linear Equations
                                                6.8 Linear Inequalities in Two Variables and Systems of Linear Inequalities
                                                6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula
                                                6.10 Functions and Their Graphs

                                                7. The Metric System
                                                7.1 Basic Terms and Conversions Within the Metric System
                                                7.2 Length, Area, and Volume
                                                7.3 Mass and Temperature
                                                7.4 Dimensional Analysis and Conversions to and from the Metric System

                                                8. Geometry
                                                8.1 Points, Lines, Planes, and Angles
                                                8.2 Polygons
                                                8.3 Perimeter and Area
                                                8.4 Volume and Surface Area
                                                8.5 Transformational Geometry, Symmetry, and Tessellations
                                                8.6 Topology
                                                8.7 Non-Euclidean Geometry and Fractal Geometry

                                                9. Mathematical Systems
                                                9.1 Groups
                                                9.2 Finite Mathematical Systems
                                                9.3 Modular Arithmetic 9.4 Matrices

                                                10. Consumer Mathematics
                                                10.1 Percent
                                                10.2 Personal Loans and Simple Interest
                                                10.3 Compound Interest
                                                10.4 Installment Buying
                                                10.5 Buying a House with a Mortgage
                                                10.6 Ordinary Annuities, Sinking Funds, and Retirement Investments

                                                11. Probability
                                                11.1 Empirical and Theoretical Probabilities
                                                11.2 Odds
                                                11.3 Expected Value (Expectation)
                                                11.4 Tree Diagrams 678
                                                11.5 OR and AND Problems
                                                11.6 Conditional Probability
                                                11.7 The Fundamental Counting Principle and Permutations
                                                11.8 Combinations
                                                11.9 Solving Probability Problems by Using Combinations
                                                11.10 Binomial Probability Formula

                                                12. Statistics
                                                12.1 Sampling Techniques and Misuses of Statistics
                                                12.2 Frequency Distributions and Statistical Graphs
                                                12.3 Measures of Central Tendency and Position
                                                12.4 Measures of Dispersion
                                                12.5 The Normal Curve
                                                12.6 Linear Correlation and Regression

                                                13. Graph Theory
                                                13.1 Graphs, Paths, and Circuits
                                                13.2 Euler Paths and Euler Circuits
                                                13.3 Hamilton Paths and Hamilton Circuits
                                                13.4 Trees 864 14 Voting and Apportionment
                                                14. Voting and Apportionment
                                                14.1 Voting Methods
                                                14.2 Flaws of the Voting Methods
                                                14.3 Apportionment Methods
                                                14.4 Flaws of the Apportionment Methods
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A Survey of Mathematics with Applications, 11th Edition © 2021
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