Purely inseparable extension: Difference between revisions

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{{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''⊆''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''] = ''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''] = ''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.