section_path: Contributions to mathematics and physics/First epoch (1908–1919)/Algebraic invariant theory
section_name: Algebraic invariant theory
paragraph_text: Algebraic invariant theory
  
  Much of Noether's work in the first epoch of her career was associated with invariant theory, principally algebraic invariant theory. Invariant theory is concerned with expressions that remain constant (invariant) under a group of transformations. As an everyday example, if a rigid yardstick is rotated, the coordinates (x, y, z) and (x, y, z) of its endpoints change, but its length L given by the formula L = Δ x + Δ y + Δ z remains the same. Invariant theory was an active area of research in the later nineteenth century, prompted in part by Felix Klein 's Erlangen program, according to which different types of geometry should be characterized by their invariants under transformations, e.g., the cross-ratio of projective geometry.
  
  An example of an invariant is the discriminant B − 4 AC of a binary quadratic form x· A x + y· B x + y· C y, where x and y are vectors and " · " is the dot product or " inner product " for the vectors. A, B, and C are linear operators on the vectors – typically matrices.
  
  The discriminant is called "invariant" because it is not changed by linear substitutions x → a x + b y and y → c x + d y with determinant ad − bc = 1. These substitutions form the special linear group SL.
  
  One can ask for all polynomials in A, B, and C that are unchanged by the action of SL; these are called the invariants of binary quadratic forms and turn out to be the polynomials in the discriminant.
  
  More generally, one can ask for the invariants of homogeneous polynomials A x y +... + A x y of higher degree, which will be certain polynomials in the coefficients A,..., A, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.
  
  One of the main goals of invariant theory was to solve the " finite basis problem ". The sum or product of any two invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting with a finite list of invariants, called generators, and then, adding or multiplying the generators together. For example, the discriminant gives a finite basis (with one element) for the invariants of binary quadratic forms.
  
  Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables. He proved this by giving a constructive method for finding all of the invariants and their generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890, David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of variables. Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the special orthogonal group.