Editing Field extension (section)

== Normal, separable and Galois extensions ==
An algebraic extension <math>L/K</math> is called [[normal extension|normal]] if every [[irreducible polynomial]] in ''K''[''X''] that has a root in ''L'' completely factors into linear factors over ''L''. Every algebraic extension ''F''/''K'' admits a normal closure ''L'', which is an extension field of ''F'' such that <math>L/K</math> is normal and which is minimal with this property.

An algebraic extension <math>L/K</math> is called [[separable extension|separable]] if the minimal polynomial of every element of ''L'' over ''K'' is [[separable polynomial|separable]], i.e., has no repeated roots in an algebraic closure over ''K''. A [[Galois extension]] is a field extension that is both normal and separable.

A consequence of the [[primitive element theorem]] states that every finite separable extension has a primitive element (i.e. is simple).

Given any field extension <math>L/K</math>, we can consider its '''automorphism group''' <math>\text{Aut}(L/K)</math>, consisting of all field [[automorphism]]s ''α'': ''L'' → ''L'' with ''α''(''x'') = ''x'' for all ''x'' in ''K''. When the extension is Galois this automorphism group is called the [[Galois group]] of the extension. Extensions whose Galois group is [[abelian group|abelian]] are called [[abelian extension]]s.

For a given field extension <math>L/K</math>, one is often interested in the intermediate fields ''F'' (subfields of ''L'' that contain ''K''). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a [[bijection]] between the intermediate fields and the [[subgroup]]s of the Galois group, described by the [[fundamental theorem of Galois theory]].