Editing Separable extension (section)

==Separable extensions within algebraic extensions== 
Let <math>E \supseteq F</math> be an algebraic extension of fields of characteristic {{math|''p''}}. The separable closure of {{math|''F''}} in {{math|''E''}} is <math>S=\{\alpha\in E \mid \alpha \text{ is separable over } F\}.</math> For every element <math>x\in E\setminus S</math> there exists a positive integer {{math|''k''}} such that <math>x^{p^k}\in S,</math> and thus {{math|''E''}} is a [[purely inseparable extension]] of  {{math|''S''}}. It follows that  {{math|''S''}} is the unique intermediate field that is ''separable'' over {{math|''F''}} and over which {{math|''E''}} is ''purely inseparable''.<ref>Isaacs, Theorem 19.14, p. 300</ref>

If <math>E \supseteq F</math> is a [[finite extension]], its [[degree of a field extension|degree]]  {{math|[''E'' : ''F'']}} is the product of the degrees {{math|[''S'' : ''F'']}} and  {{math|[''E'' : ''S'']}}. The former, often denoted {{math|[''E'' : ''F'']<sub>sep</sub>}}, is referred to as the ''separable part'' of  {{math|[''E'' : ''F'']}},  or as the '''{{visible anchor|separable degree}}''' of {{math|''E''/''F''}}; the latter is referred to as the ''inseparable part'' of the degree or the '''{{visible anchor|inseparable degree}}'''.<ref name="Isaacs302">Isaacs, p. 302</ref> The inseparable degree is 1 in characteristic zero and a power of {{math|''p''}} in characteristic {{math|''p'' > 0}}.<ref>{{harvnb|Lang|2002|loc=Corollary V.6.2}}</ref>

On the other hand, an arbitrary algebraic extension <math>E\supseteq F</math> may not possess an intermediate extension {{math|''K''}} that is ''purely inseparable'' over {{math|''F''}} and over which {{math|''E''}} is ''separable''. However, such an intermediate extension may exist if, for example, <math>E\supseteq F</math> is a finite degree normal extension (in this case, {{math|''K''}} is the fixed field of the Galois group of {{math|''E''}} over {{math|''F''}}). Suppose that such an intermediate extension does exist, and {{math|[''E'' : ''F'']}} is finite, then {{math|1=[''S'' : ''F''] = [''E'' : ''K'']}}, where {{math|''S''}} is the separable closure of {{math|''F''}} in  {{math|''E''}}.<ref>Isaacs, Theorem 19.19, p. 302</ref> The known proofs of this equality use the fact that if <math>K\supseteq F</math> is a purely inseparable extension, and if {{math|''f''}} is a separable irreducible polynomial in {{math|''F''[''X'']}}, then {{math|''f''}} remains irreducible in ''K''[''X'']<ref>Isaacs, Lemma 19.20, p. 302</ref>). This equality implies that, if {{math|[''E'' : ''F'']}} is finite, and {{math|''U''}} is an intermediate field between {{math|''F''}} and {{math|''E''}}, then {{math|1=[''E'' : ''F'']<sub>sep</sub> = [''E'' : ''U'']<sub>sep</sub>⋅[''U'' : ''F'']<sub>sep</sub>}}.<ref>Isaacs, Corollary 19.21, p. 303</ref>

The separable closure {{math|''F''<sup>sep</sup>}} of a field {{math|''F''}} is the separable closure of {{math|''F''}} in an [[algebraic closure]] of {{math|''F''}}. It is the maximal [[Galois extension]] of {{math|''F''}}.  By definition, {{math|''F''}} is [[perfect field|perfect]] if and only if its separable and algebraic closures coincide.