Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable;^[1] 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.^[2]

The property of an extension being Galois behaves well with respect to field composition and intersection.^[3]

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