In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.
Contents
For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.
Definition
Suppose that E is an extension of the field F (written as E/F and read E over F). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism α from E to E such that α(x) = x for each x in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by Aut(E/F).
If E/F is a Galois extension, then Aut(E/F) is called the Galois group of (the extension) E over F, and is usually denoted by Gal(E/F).
If E/F is not a Galois extension, then the Galois group of (the extension) E over F is sometimes defined as Aut(G/F), where G is the Galois closure of E.
Examples
In the following examples F is a field, and C, R, Q are the fields of complex, real, and rational numbers, respectively. The notation F(a) indicates the field extension obtained by adjoining an element a to the field F.
Properties
The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology) subgroups of the Galois group correspond to the intermediate fields of the field extension.
If E/F is a Galois extension, then Gal(E/F) can be given a topology, called the Krull topology, that makes it into a profinite group.