Editing Separable extension (section)

{{Short description|Type of algebraic field extension}}
In [[field theory (mathematics)|field theory]], a branch of [[algebra]], an [[algebraic field extension]] <math>E/F</math> is called a '''separable extension''' if for every <math>\alpha\in E</math>, the [[minimal polynomial (field theory)|minimal polynomial]] of <math>\alpha</math> over {{mvar|F}} is a [[separable polynomial]] (i.e., its [[formal derivative]] is not the zero [[polynomial]], or equivalently it has no repeated [[zero of a function|root]]s in any extension field).<ref name="Isaacs281">Isaacs, p. 281</ref>  There is also a more general definition that applies when {{mvar|E}} is not necessarily algebraic over {{mvar|F}}.  An extension that is not separable is said to be ''inseparable''.

Every algebraic extension of a [[field (mathematics)|field]] of [[characteristic (algebra)#Case of fields|characteristic]] zero is separable, and every algebraic extension of a [[finite field]] is separable.<ref name="Isaacs18.11p281">Isaacs, Theorem 18.11, p. 281</ref>
It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the [[fundamental theorem of Galois theory]] is a theorem about [[normal extension]]s, which remains true in non-zero characteristic only if the extensions are also assumed to be separable.<ref>Isaacs, Theorem 18.13, p. 282</ref>

The opposite concept, a [[purely inseparable extension]], also occurs naturally, as every algebraic extension may be decomposed uniquely as a purely inseparable extension of  a separable extension. An algebraic extension <math>E/F</math> of fields of non-zero characteristic {{math|''p''}} is a purely inseparable extension if and only if for every <math>\alpha\in E\setminus F</math>, the minimal polynomial of <math>\alpha</math> over {{math|''F''}} is ''not'' a separable polynomial, or, equivalently, for every element {{math|''x''}} of {{math|''E''}}, there is a positive [[integer]] {{math|''k''}} such that <math>x^{p^k} \in F</math>.<ref name="Isaacs298">Isaacs, p. 298</ref>

The simplest nontrivial example of a (purely) inseparable extension is <math>E=\mathbb{F}_p(x) \supseteq F=\mathbb{F}_p(x^p)</math>, fields of [[rational function]]s in the indeterminate ''x'' with coefficients in the [[finite field]] <math>\mathbb{F}_p=\mathbb{Z}/(p)</math>. The element <math>x\in E</math> has minimal polynomial <math>f(X)=X^p -x^p \in F[X]</math>, having <math>f'(X) = 0</math> and a ''p''-fold multiple root, as <math>f(X)=(X-x)^p\in E[X]</math>. This is a [[simple field extension|simple]] algebraic extension of degree ''p'', as <math>E = F[x]</math>, but it is not a normal extension since the [[Galois group]] <math>\text{Gal}(E/F)</math> is [[trivial group|trivial]].