Purely inseparable extension: Difference between revisions

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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>

1.# ''E'' is purely inseparable over ''F.''

# For each element <math>\alpha\in E</math>, there exists <math>n\geq 0</math> such that <math>\alpha^{p^n}\in F</math>.

2.# For eachEach element of ''E'' has minimal polynomial over ''F'' of the form <math>\alpha\in EX^{p^n}-a</math>, therefor existssome integer <math>n\geq 0</math> suchand thatsome element <math>\alpha^{p^n}a\in F</math>.

3. Each element of ''E'' has minimal polynomial over ''F'' of the form <math>X^{p^n}-a</math> for some integer <math>n\geq 0</math> and some element <math>a\in F</math>.

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.)