# 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. 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 $E/F,$  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 $F.$
• $|\!\operatorname {Aut} (E/F)|=[E:F],$  that is, the number of automorphisms equals the degree of the extension.

Other equivalent statements are:

• Every irreducible polynomial in $F[x]$  with at least one root in $E$  splits over $E$  and is separable.
• $|\!\operatorname {Aut} (E/F)|\geq [E:F],$  that is, the number of automorphisms is at least the degree of the extension.
• $F$  is the fixed field of a subgroup of $\operatorname {Aut} (E).$
• $F$  is the fixed field of $\operatorname {Aut} (E/F).$
• There is a one-to-one correspondence between subfields of $E/F$  and subgroups of $\operatorname {Aut} (E/F).$

## Examples

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

• Take any field $E$ , any finite subgroup of $\operatorname {Aut} (E)$ , and let $F$  be the fixed field.
• Take any field $F$ , any separable polynomial in $F[x]$ , and let $E$  be its splitting field.

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

An algebraic closure ${\bar {K}}$  of an arbitrary field $K$  is Galois over $K$  if and only if $K$  is a perfect field.