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. Properties If the characteristic of a field F is a (non-zero) prime number p, and if is a purely inseparable extension, then if , K is purely inseparable over F and E is purely inseparable over K. Furthermore, if [E : F] is finite, then it is a power of p, the characteristic of F. Conversely, if is such that and are purely inseparable extensions, then E is purely inseparable over F. An algebraic extension is an inseparable extension if and only if there is some such that the minimal polynomial of over F is not a separable polynomial (i.e., an algebraic extension is inseparable if and only if it is not separable; note, however, that an inseparable extension is not the same thing as a purely inseparable extension). If is a finite degree non-trivial inseparable extension, then [E : F] is necessarily divisible by the characteristic of F. If is a finite degree normal extension, and if , then K is purely inseparable over F and E is separable over K. Galois correspondence for purely inseparable extensions 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. A purely inseparable 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 . and gave an extension