Editing Galois extension (section)

==Characterization of Galois extensions==
An important theorem of [[Emil Artin]] states that for a [[finite extension]] <math>E/F,</math> each of the following statements is equivalent to the statement that <math>E/F</math> is Galois:

*<math>E/F</math> is a [[normal extension]] and a [[separable extension]].
*<math>E</math> is a [[splitting field]] of a [[separable polynomial]] with coefficients in <math>F.</math>
*<math>|\!\operatorname{Aut}(E/F)| = [E:F],</math> that is, the number of automorphisms equals the [[degree (field theory)|degree]] of the extension.

Other equivalent statements are:

*Every irreducible polynomial in <math>F[x]</math> with at least one root in <math>E</math> splits over <math>E</math> and is separable.
*<math>|\!\operatorname{Aut}(E/F)| \geq [E:F],</math> that is, the number of automorphisms is at least the degree of the extension.
*<math>F</math> is the fixed field of a subgroup of <math>\operatorname{Aut}(E).</math>
*<math>F</math> is the fixed field of <math>\operatorname{Aut}(E/F).</math>
*There is a one-to-one [[Fundamental theorem of Galois theory#Explicit description of the correspondence|correspondence]] between subfields of <math>E/F</math> and subgroups of <math>\operatorname{Aut}(E/F).</math>

An infinite field extension <math>E/F</math> is Galois if and only if <math>E</math> is the union of finite Galois subextensions <math>E_i/F</math> indexed by an (infinite) index set <math>I</math>, i.e. <math>E=\bigcup_{i\in I}E_i</math> and the Galois group is an [[inverse limit]] <math>\operatorname{Aut}(E/F)=\varprojlim_{i\in I}{\operatorname{Aut}(E_i/F)}</math> where the inverse system is ordered by field inclusion  <math>E_i\subset E_j</math>.{{sfn|Milne|2022|p=102|loc=example 7.26}}