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.
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