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==Purely inseparable extensions==
An algebraic extension <math>E\supseteq F</math> 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 ''F'' is ''not'' a [[separable polynomial]].<ref name="Isaacs298">Isaacs, p. 298</ref> If ''F'' is any field, the trivial extension <math>F\supseteq F</math> is purely inseparable; for the field ''F'' to possess a ''non-trivial'' purely inseparable extension, it must be imperfect as outlined in the above section.
Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known. If <math>E\supseteq F</math> is an algebraic extension with (non-zero) prime characteristic ''p'', then the following are equivalent:<ref>Isaacs, Theorem 19.10, p. 298</ref>
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It follows from the above equivalent characterizations that if <math>E=F[\alpha]</math> (for ''F'' a field of prime characteristic) such that <math>\alpha^{p^n}\in F</math> for some integer <math>n\geq 0</math>, then ''E'' is purely inseparable over ''F''.<ref>Isaacs, Corollary 19.11, p. 298</ref> (To see this, note that the set of all ''x'' such that <math>x^{p^n}\in F</math> for some <math>n\geq 0</math> forms a field; since this field contains both <math>\alpha</math> and ''F'', it must be ''E'', and by condition 2 above, <math>E\supseteq F</math> must be purely inseparable.)
If ''F'' is an imperfect field of prime characteristic ''p'', choose <math>a\in F</math> such that ''a'' is not a ''p''th power in ''F'', and let ''f''(''X'')=''X''<sup>p</sup>−''a''. Then ''f'' has no root in ''F'', and so if ''E'' is a splitting field for ''f'' over ''F'', it is possible to choose <math>\alpha</math> with <math>f(\alpha)=0</math>. In particular, <math>\alpha^{p}=a</math> and by the property stated in the paragraph directly above, it follows that <math>F[\alpha]\supseteq F</math> is a non-trivial purely inseparable extension (in fact, <math>E=F[\alpha]</math>, and so <math>E\supseteq F</math> is automatically a purely inseparable extension).<ref name="Isaacs299">Isaacs, p. 299</ref>
Purely inseparable extensions do occur naturally; for example, they occur in [[algebraic geometry]] over fields of prime characteristic. If ''K'' is a field of characteristic ''p'', and if ''V'' is an [[algebraic variety]] over ''K'' of dimension greater than zero, the [[function field of an algebraic variety|function field]] ''K''(''V'') is a purely inseparable extension over the [[subfield]] ''K''(''V'')<sup>''p''</sup> of ''p''th powers (this follows from condition 2 above). Such extensions occur in the context of multiplication by ''p'' on an [[elliptic curve]] over a finite field of characteristic ''p''.
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