Galois extension

Galois extension

In mathematics, a Galois extension is a field extension that has a Galois group. A fundamental result of Galois theory characterises these extensions: a finite extension of fields L/K is a Galois extension if and only if it is both a normal extension and a separable extension.

This criterion can be used in practice to show that extensions have Galois groups. It states, in more concrete terms, that L is built up from K as a compositum of a number of splitting fields of separable polynomials.

There is also the result of Emil Artin that starts with L given. If G is a finite group of automorphisms of L, then the fixed field K of G is such that L/K is a Galois extension.