 Galois Extension
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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 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 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.