# Galois 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*. One says that such an extension is **Galois**. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. ^{[1]}

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 *E*/*F*, each of the following statements is equivalent to the statement that *E*/*F* is Galois:

*E*/*F*is a normal extension and a separable extension.*E*is a splitting field of a separable polynomial with coefficients in*F*.- [
*E*:*F*] = |Aut(*E*/*F*)|; that is, the degree of the field extension is equal to the order of the automorphism group of*E*/*F*.

## Examples

Adjoining to the rational number field the square root of 2 gives a Galois extension, while adjoining the cube 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 *X*^{2} − 2; the second has normal closure that includes the complex cube 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 *X*^{3} − 2 has just one real root.

An algebraic closure of an arbitrary field is Galois over if and only if is a perfect field.

## References

- ↑ See the article Galois group for definitions of some of these terms and some examples.

## See also

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