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Hi, Let $A$ be a division algebra over a local field $F$ of dimension $n^2$ and $K$ be an extension of $F$ of degree $n$. Then if follows from COROLLARY 2 in page 225 of Weil's Basic Number Theory that $A$ is split over $K$. Now my question is

  1. Does $A$ contain a subfield isomorphic to $K$ ? Why?

  2. Could you descirbe the general picture about "maximal subfields in central simple algebras" over a (not necessarily local) field ? Or tell me some references.

Thank you !

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3 Answers 3

up vote 7 down vote accepted

Corollaries 3.4 and 3.7 on pages 130 and 131 of Milne's notes (jmilne.org/math/CourseNotes/CFT.pdf) will give you the answers.

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Thanks. So any finite extension $K$ of a local field $F$ of degree $n$ could be embedded into the division algebra $A$ over $F$ of degree $n^2$. Is it right ? –  user4245 Aug 1 '12 at 13:35
And in the case of global fields, when could the field $K$ be embedded into the division algebra $A$ over $F$ ? One expect $K$ and $A$ should be "compatible" locally. –  user4245 Aug 1 '12 at 16:27

To address 2.: for any central simple algebra $A$ over a field $k$, there is a well-developed theory describing the relations between finite splitting fields $l/k$ for $A$ and fields which are sub-$k$-algebras of $A$. The most important result is probably this one:

Theorem: Let $A$ be a central simple algebra over a field $k$, of dimension $n^2$. For a field extension $l/k$ of degree $n$, the following are equivalent:
(i) There is a $k$-algebra embedding $l \hookrightarrow A$.
(ii) $l$ is a splitting field for $A$.

A proof of this result can be found, for instance, in $\S 6.2$ of these notes on noncommutative algebra. Citations to more substantial treatments are given in $\S 6.10$.

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Thanks very much for this nice reference for central simple algebra. I am a bit confused: don't you need to pass to an suitable $A^\prime$ similar to $A$ in (i) of the theorem ? –  user4245 Aug 2 '12 at 6:33
@unknown: I don't think so (and the statement and proof of the result I quote appears in my notes). Why do you think you need to pass to $A'$? –  Pete L. Clark Aug 2 '12 at 7:11
Oh,I see! I just miss the condition "of dimension $n^2$" in your statement. Sorry and thanks again. –  user4245 Aug 2 '12 at 7:18

The answer to your first question can also be found in Weil's Basic Number Theory. See Corollary 3 on page 180, and note that if two central simple algebras of the same dimension are similar then they are isomorphic.

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Thank you so much! I just miss it. –  user4245 Aug 2 '12 at 6:26
Note that Weil restricts to separable-extension case for simplicity. –  user4245 Aug 2 '12 at 12:06

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