Emmy Noether: Mathematician Trailblazer (Part VI)

First epoch (1908–1919): 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 (x1, y1, z1) and (x2, y2, z2) of its endpoints change, but its length L given by the formula L2 = Δx2 + Δy2 + Δz2 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 B2 − 4 A C 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, y → c x + d y with determinant a d − b c = 1. These substitutions form the special linear group SL2.

One can ask for all polynomials in A, B, and C that are unchanged by the action of SL2; 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 A0 xr y0 + ... + Ar x0 yr of higher degree, which will be certain polynomials in the coefficients A0, ..., Ar, 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.

First epoch (1908–1919): Galois Theory

Galois theory concerns transformations of number fields that permute the roots of an equation. Consider a polynomial equation of a variable x of degree n, in which the coefficients are drawn from some ground field, which might be, for example, the field of real numbers, rational numbers, or the integers modulo 7. There may or may not be choices of x, which make this polynomial evaluate to zero.

Such choices, if they exist, are called roots. If the polynomial is x2 + 1 and the field is the real numbers, then the polynomial has no roots, because any choice of x makes the polynomial greater than or equal to one. If the field is extended, however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree.

Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, +i and −i, where i is the imaginary unit, that is, i 2 = −1.  More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial.

The Galois group of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial. (In mathematical jargon, these transformations are called automorphisms.) The Galois group of x2 + 1 consists of two elements: The identity transformation, which sends every complex number to itself, and complex conjugation, which sends +i to −i. Since the Galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. Each root can move to another root, however, so transformation determines a permutation of the n roots among themselves. The significance of the Galois group derives from the fundamental theorem of Galois Theory, which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the subgroups of the Galois group.

In 1918, Noether published a paper on the inverse Galois problem.  Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to "Noether's problem", which asks whether the fixed field of a subgroup G of the permutation group Sn acting on the field k(x1, ... , xn) always is a pure transcendental extension of the field k. (She first mentioned this problem in a 1913 paper, where she attributed the problem to her colleague Fischer.) She showed this was true for n = 2, 3, or 4. In 1969, R.G. Swan found a counter-example to Noether's problem, with n = 47 and G a cyclic group of order 47.  (Although this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem remains unsolved.