Purely inseparable extension: Difference between revisions

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In algebra, a '''Purelypurely inseparable extension''' of fields is an extension ''k''⊆''K'' of fields of characteristic ''p''>0 such that every element of ''K'' is a root of an equation of the form ''x''^''q'' = ''a'', with ''q'' a power of ''p'' and ''a'' in ''k''. Purely inseparable extensions are sometimes called '''radicial extensions''', which should not be confused with the similar-sounding but more general notion of [[radical extension]]s.

==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"/> 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.