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==Jacobson's Galois theory 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|restricted]]) Lie algebras 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''<sup>''n''</sup>, 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''
There are extensions of this result to purely inseparable extensions of larger exponent, where derivations are replaced by higher derivations.
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