# Galois extension – Wikipedia

Algebraic field extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.[a]

A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.

## Characterization of Galois extensions

An important theorem of Emil Artin states that for a finite extension

${displaystyle E/F,}$ each of the following statements is equivalent to the statement that

${displaystyle E/F}$ is Galois:

Other equivalent statements are:

## Examples

There are two basic ways to construct examples of Galois extensions.

Adjoining to the rational number field the square root of 2 gives a Galois extension, while adjoining the cubic root of 2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The first of them is the splitting field of

${displaystyle x^{2}-2}$ ; the second has normal closure that includes the complex cubic roots of unity, and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and

${displaystyle x^{3}-2}$ has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.

An algebraic closure

${displaystyle {bar {K}}}$ of an arbitrary field

${displaystyle K}$ is Galois over

${displaystyle K}$ if and only if

${displaystyle K}$ is a perfect field.

1. ^ See the article Galois group for definitions of some of these terms and some examples.