Purely inseparable extension: Difference between revisions

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In [[abstract algebra|algebra]], a '''purely inseparable extension''' of fields[[Field (mathematics)|field]]s is an extension ''k'' ⊆ ''K'' of fields of [[characteristic of a field|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 (field theory)|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>

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

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

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

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'')&nbsp;=&nbsp;''X''^''p''&nbsp;&minus;&nbsp;''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>

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>.

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 [[Field extension|subfield]] ''K''(''V'')^''p'' 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''.

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'')&nbsp;=&nbsp;''X''^p&nbsp;&minus;&nbsp;''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'')^''p'' 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''.

===Properties===

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==Galois correspondence for purely inseparable extensions==

{{harvs|txt|last=Jacobson|year1=1937|year2=1944}} introduced a variation of [[Galois theory]] for purely inseparable extensions of exponent 1, where the Galois groups of field automorphisms in Galois theory are replaced by [[restricted Lie algebra]]s of derivations. The simplest case is for finite index purely inseparable extensions ''K''&sube;''L'' of exponent at most 1 (meaning that the ''p''th power of every element of ''L'' is in ''K''). In this case the Lie algebra of ''K''-derivations of ''L'' is a restricted Lie algebra that is also a vector space of dimension ''n'' over ''L'', where [''L'':''K'']&nbsp;=&nbsp;''p''^''n'', and the intermediate fields in ''L'' containing ''K'' correspond to the restricted Lie subalgebras of this Lie algebra that are vector spaces over ''L''. Although the Lie algebra of derivations is a vector space over ''L'', it is not in general a Lie algebra over ''L'', but is a Lie algebra over ''K'' of dimension ''n''[''L'':''K'']&nbsp;=&nbsp;''np''^''n''.

A purely separableinseparable extension is called a '''modular extension''' if it is a tensor product of simple extensions, so in particular every extension of exponent 1 is modular, but there are non-modular extensions of exponent 2 {{harv|Weisfeld|1965}}.

{{harvtxt|Sweedler|1968}} and {{harvtxt|Gerstenhaber|Zaromp|1970}} gave an extension of the Galois correspondence to modular purely inseparable extensions, where derivations are replaced by higher derivations.

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{{reflist}}

*{{Citation | last1=Gerstenhaber | first1=Murray |authorlink1=Murray Gerstenhaber| last2=Zaromp | first2=Avigdor | title=On the Galois theory of purely inseparable field extensions | doi=10.1090/S0002-9904-1970-12535-6 | idmr={{MR|0266904}} | year=1970 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=76 | issue=5 | pages=1011–1014| doi-access=free }}

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| isbn = 0-534-19002-2

}}

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*{{Citation | last1=Weisfeld | first1=Morris | title=Purely inseparable extensions and higher derivations | url=http://www.jstor.org/stable/=1994126 | idmr={{MR|0191895}} | year=1965 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=116 | pages=435–449 | doi=10.2307/1994126| doi-access=free }}

[[Category:Field theory(mathematics)]]