Galois extension

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In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions (described below); one also says that the 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.

The definition is as follows. An algebraic field extension E/F is Galois if it is normal and separable. Equivalently, the extension E/F is Galois if and only if it is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. (See the article Galois group for definitions of some of these terms and some examples.)

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, then E/F is a Galois extension, where F is the fixed field of G.

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 the 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.