Editing Galois theory (section)

==Inseparable extensions==
In the form mentioned above, including in particular the [[fundamental theorem of Galois theory]], the theory only considers Galois extensions, which are in particular separable. General field extensions can be split into a separable, followed by a [[purely inseparable field extension]]. For a purely inseparable extension ''F'' / ''K'', there is a Galois theory where the Galois group is replaced by the [[vector space]] of [[derivation (differential algebra)|derivations]], <math>Der_K(F, F)</math>, i.e., ''K''-[[linear map|linear endomorphisms]] of ''F'' satisfying the Leibniz rule. In this correspondence, an intermediate field ''E'' is assigned <math>Der_E(F, F) \subset Der_K(F, F)</math>. Conversely, a [[linear subspace|subspace]] <math>V \subset Der_K(F, F)</math> satisfying appropriate further conditions is mapped to <math>\{x \in F, f(x)=0\ \forall f \in V\}</math>. Under the assumption <math>F^p \subset K</math>, {{harvtxt|Jacobson|1944}} showed that this establishes a one-to-one correspondence. The condition imposed by Jacobson has been removed by {{harvtxt|Brantner|Waldron|2020}}, by giving a correspondence using notions of [[derived algebraic geometry]].