In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation of the form xq = 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 extensions.
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.
is an algebraic extension with (non-zero) prime characteristic p, then the following are equivalent:[2] It follows from the above equivalent characterizations that if
, then E is purely inseparable over F.[3] (To see this, note that the set of all x such that
If F is an imperfect field of prime characteristic p, choose
Then f has no root in F, and so if E is a splitting field for f over F, it is possible to choose
and by the property stated in the paragraph directly above, it follows that
is a non-trivial purely inseparable extension (in fact,
is automatically a purely inseparable extension).
[4] 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 K(V) is a purely inseparable extension over the subfield K(V)p of pth 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. Jacobson (1937, 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 algebras of derivations.
The simplest case is for finite index purely inseparable extensions K⊆L of exponent at most 1 (meaning that the pth 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] = pn, 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] = npn.
Sweedler (1968) and Gerstenhaber & Zaromp (1970) gave an extension of the Galois correspondence to modular purely inseparable extensions, where derivations are replaced by higher derivations.