Purely inseparable extension: Difference between revisions

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===Properties===

*If the characteristic of a field ''F'' is a (non-zero) prime number ''p'', and if <math>E\supseteq F</math> is a purely inseparable extension, then if <math>F\subseteq K\subseteq E</math>, ''K'' is purely inseparable over ''F'' and ''E'' is purely inseparable over ''K''. Furthermore, if [''E'' : ''F''] is finite, then it is a power of ''p'', the characteristic of ''F''.<ref>Isaacs, Corollary 19.12, p. 299</ref>

*Conversely, if <math>F\subseteq K\subseteq E</math> is such that <math>F\subseteq K</math> and <math>K\subseteq E</math> are purely inseparable extensions, then ''E'' is purely inseparable over ''F''.<ref>Isaacs, Corollary 19.13, p. 300</ref>

*An algebraic extension <math>E\supseteq F</math> is an '''inseparable extension''' if and only if there is ''some'' <math>\alpha\in E\setminus F</math> such that the minimal polynomial of <math>\alpha</math> over ''F'' is ''not'' a [[separable polynomial]] (i.e., an algebraic extension is inseparable if and only if it is not separable; note, however, that an inseparable extension is not the same thing as a purely inseparable extension). If <math>E\supseteq F</math> is a finite degree non-trivial inseparable extension, then [''E'' : ''F''] is necessarily divisible by the characteristic of ''F''.<ref>Isaacs, Corollary 19.16, p. 301</ref>

*If <math>E\supseteq F</math> is a finite degree normal extension, and if <math>K=\mbox{Fix}(\mbox{Gal}(E/F))</math>, then ''K'' is purely inseparable over ''F'' and ''E'' is separable over ''K''.<ref>Isaacs, Theorem 19.18, p. 301</ref>

==Jacobson's Galois theory for purely inseparable extensions==

Jacobson found a variation of Galois theory for purely inseparable extensions, where the Galois groups of field automorphisms in Galois theory are replaced by (restricted) Lie algebras of derivations.

==References==