==Jacobson's Galois theory for purely inseparable extensions==
{{harvtxtharvs|txt|last=Jacobson|year1=1937|year2=1944}} foundintroduced 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 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 of dimension ''n'', 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. There are extensions of this to purely inseparable extensions of larger exponent, where derivations are replaced by higher derivations.
