Some remarks on higher derivations of finite rank in a field of a positive characteristic

Abstract

In this short note we give a generalization of an approximation theorem on iterated higher derivations given by F. K. Schmidt in a paper [_2J (see Satz 14). Our generalization is done by determining all the iterated higher derivations of finite rank in any field K of a positive characteristic p. The following result on a derivation d in K will play an essential role in the proof: if we have d = 0, then d~\a) = 0 if and only if a = d(β) for some β in K*\ We shall give a proof of this fact using the Jacobson-Bourbaki's theorem which asserts the existence of a 1 — 1 correspondence between subfields of finite codimension in a field K and certain subrings of the ring M(K) of endomorphisms of the additive group (K, +) . Lastly we shall be concerned with conditions for a purely inseparable extension K of finite degree over a field k to be a tensor product of simple extensions over k. These conditions will be given in terms of higher derivations in K.