Editing Galois extension (section)

{{Short description|Algebraic field extension}}
In [[mathematics]], a '''Galois extension''' is an [[Algebraic extension|algebraic]] [[field extension]] ''E''/''F'' that is [[normal extension|normal]] and [[separable extension|separable]];{{sfn|Lang|2002|p=262}} or equivalently, ''E''/''F'' is algebraic, and the [[Fixed field|field fixed]] by the [[automorphism group]] Aut(''E''/''F'') is precisely the base [[Field (mathematics)|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]].{{efn|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'' with fixed field ''F'', then ''E''/''F'' is a Galois extension.{{sfn|Lang|2002|p=264|loc=Theorem 1.8}}

The property of an extension being Galois behaves well with respect to [[Composite field (mathematics)| field composition and intersection]].{{sfn|Milne|2022|p=40f|loc=ch. 3 and 7}}