section_path: Contributions to mathematics and physics/First epoch (1908–1919)/Galois theory
section_name: Galois theory
paragraph_text: 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. For example, if the polynomial is x + 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 = −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. (These transformations are called automorphisms.) The Galois group of x + 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 S acting on the field k (x,..., x) 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, Richard 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.