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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. The extension ''E''/''F'' is Galois if 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''.

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Characterization of Galois extensions

Characterization of Galois extensions

An important theorem of Emil Artin states that a finite extension ''E''/''F'' is Galois if and only if any one of the following conditions holds:

★ ''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.