Editing Separable extension (section)

==Separable elements and separable extensions==
Let <math>E\supseteq F</math> be a field extension. An element <math>\alpha\in E</math> is '''separable''' over {{math|''F''}} if it is algebraic over {{math|''F''}}, and its [[minimal polynomial (field theory)|minimal polynomial]] is separable (the minimal polynomial of an element is necessarily irreducible).

If <math>\alpha,\beta\in E</math> are separable over {{math|''F''}}, then <math>\alpha+\beta</math>, <math>\alpha\beta</math> and <math>1/\alpha</math> are separable over ''F''.

Thus the set of all elements in {{math|''E''}} separable over {{math|''F''}} forms a subfield of {{math|''E''}}, called the '''separable closure''' of {{math|''F''}} in {{math|''E''}}.<ref>Isaacs, Lemma 19.15, p. 300</ref>

The separable closure of {{math|''F''}} in an [[algebraic closure]] of {{math|''F''}} is simply called the '''[[separable closure]]''' of {{math|''F''}}. Like the algebraic closure, it is unique up to an isomorphism, and in general, this isomorphism is not unique.

A field extension <math>E\supseteq F</math> is '''separable''', if {{math|''E''}} is the separable closure of {{math|''F''}} in {{math|''E''}}. This is the case if and only if {{math|''E''}} is generated over {{math|''F''}} by separable elements.

If <math>E\supseteq L\supseteq F</math> are field extensions, then {{math|''E''}} is separable over {{math|''F''}} if and only if {{math|''E''}} is separable over {{math|''L''}} and {{math|''L''}} is separable over {{math|''F''}}.<ref>Isaacs, Corollary 18.12, p. 281 and Corollary 19.17, p. 301</ref>

If <math>E\supseteq F</math> is a [[finite extension]] (that is {{math|''E''}} is a {{math|''F''}}-[[vector space]] of finite [[dimension (vector space)|dimension]]), then the following are equivalent.
# {{math|''E''}} is separable over {{math|''F''}}.
# <math>E = F(a_1, \ldots, a_r)</math> where <math>a_1, \ldots, a_r</math> are separable elements of {{math|''E''}}.
# <math>E = F(a)</math> where {{math|''a''}} is a separable element of {{math|''E''}}.
# If {{math|''K''}} is an algebraic closure of {{math|''F''}}, then there are exactly <math>[E : F]</math> [[field homomorphism]]s of {{math|''E''}} into {{math|''K''}} that fix {{math|''F''}}.
# For any normal extension {{math|''K''}}  of {{math|''F''}} that contains {{math|''E''}}, then there are exactly <math>[E : F]</math> field homomorphisms of {{math|''E''}} into {{math|''K''}} that fix {{math|''F''}}.
The equivalence of 3. and 1. is known as the ''[[primitive element theorem]]'' or  ''Artin's theorem on primitive elements''.
Properties 4. and 5. are the basis of [[Galois theory]], and, in particular, of the [[fundamental theorem of Galois theory]].