Editing Separable extension (section)

==Separable and inseparable polynomials==
An [[irreducible polynomial]] {{math|''f''}} in {{math|''F''[''X'']}} is [[separable polynomial|separable]] if and only if it has distinct roots in any [[field extension|extension]] of {{math|''F''}}. That is, if it is the product of distinct linear factors {{math|''X'' - ''a''}} in some [[algebraically closed field|algebraic closure]] of {{math|''F''}}.<ref>Isaacs, p. 280</ref> 
Let {{math|''f''}} in {{math|''F''[''X'']}} be an irreducible polynomial and {{math|''f'' '}} its [[formal derivative]]. Then the following are equivalent conditions for the irreducible polynomial {{math|''f''}} to be separable:
* If {{math|''E''}} is an extension of {{math|''F''}} in which {{math|''f''}} is a product of linear factors then no square of these factors divides {{math|''f''}} in {{math|''E''[''X'']}} (that is {{math|''f''}} is [[square-free polynomial|square-free]] over {{math|''E''}}).<ref name=IsaacsLem18.7>Isaacs, Lemma 18.7, p. 280</ref>
* There exists an extension {{math|''E''}} of {{math|''F''}} such that {{math|''f''}} has {{math|deg(''f'')}} pairwise distinct roots in {{math|''E''}}.<ref name=IsaacsLem18.7/>
* The constant {{math|1}} is a [[polynomial greatest common divisor]] of {{math|''f''}} and {{math|''f'' '}}.<ref>Isaacs, Theorem 19.4, p. 295</ref>
* The formal derivative {{math|''f'' '}} of {{math|''f''}} is not the zero polynomial.<ref>Isaacs, Corollary 19.5, p. 296</ref>
* Either the characteristic of  {{math|''F''}} is zero, or the characteristic is {{math|''p''}}, and {{math|''f''}} is not of the form <math>\textstyle\sum_{i=0}^k a_iX^{pi}.</math>

Since the formal derivative of a positive degree polynomial can be zero only if the field has prime characteristic, for an irreducible polynomial to not be separable, its coefficients must lie in a field of prime characteristic. More generally, an irreducible (non-zero) polynomial {{math|''f''}} in {{math|''F''[''X'']}} is not separable, if and only if the characteristic of {{math|''F''}} is a (non-zero) prime number {{math|''p''}}, and {{math|1=''f''(''X'')=''g''(''X''<sup>''p''</sup>}}) for some ''irreducible'' polynomial {{math|''g''}} in {{math|''F''[''X'']}}.<ref>Isaacs, Corollary 19.6, p. 296</ref> By repeated application of this property, it follows that in fact, <math>f(X)=g(X^{p^n})</math> for a non-negative integer {{math|''n''}} and some ''separable irreducible'' polynomial {{math|''g''}} in {{math|''F''[''X'']}} (where {{math|''F''}} is assumed to have prime characteristic ''p'').<ref>Isaacs, Corollary 19.9, p. 298</ref>

If the [[Frobenius endomorphism]] <math>x\mapsto x^p</math> of {{math|''F''}} is not surjective, there is an element <math>a\in F</math> that is not a {{math|''p''}}th power of an element of {{math|''F''}}. In this case, the polynomial <math>X^p-a</math> is irreducible and inseparable. Conversely, if there exists an inseparable irreducible (non-zero) polynomial <math>\textstyle f(X)=\sum a_iX^{ip}</math> in {{math|''F''[''X'']}}, then the [[Frobenius endomorphism]] of {{math|''F''}} cannot be an [[automorphism]], since, otherwise, we would have <math>a_i=b_i^p</math> for some <math>b_i</math>, and the polynomial {{math|''f''}} would factor as <math>\textstyle \sum a_iX^{ip}=\left(\sum b_iX^{i}\right)^p.</math><ref>Isaacs, Theorem 19.7, p. 297</ref>

If {{math|''K''}} is a finite field of prime characteristic ''p'', and if {{math|''X''}} is an [[indeterminate (variable)|indeterminate]], then the [[field of rational functions]] over {{math|''K''}}, {{math|''K''(''X'')}}, is necessarily [[Imperfect field|imperfect]], and the polynomial {{math|1=''f''(''Y'')=''Y''<sup>''p''</sup>−''X''}} is inseparable (its formal derivative in ''Y'' is 0).<ref name="Isaacs281"/> More generally, if ''F'' is any field of (non-zero) prime characteristic for which the [[Frobenius endomorphism]] is not an automorphism, ''F'' possesses an inseparable algebraic extension.<ref name="Isaacs299">Isaacs, p. 299</ref>

A field ''F'' is [[Perfect field|perfect]] if and only if all irreducible polynomials are separable. It follows that {{math|''F''}} is perfect if and only if either {{math|''F''}} has characteristic zero, or {{math|''F''}} has (non-zero) prime characteristic {{math|''p''}} and the [[Frobenius endomorphism]] of {{math|''F''}} is an automorphism. This includes every finite field.