GENERIC MODULES OF LINEAR-DIFFERENTIAL FORMS AND PICARD-VESSIOT EXTENSIONS IN ARBITRARY CHARACTERISTIC

Smith in his doctoral dissertation (1990) constructed generic polynomials for cyclic extensions of prime power degree. Goldman1 had already considered the analogous theory for Picard-Vessiot extensions of differential fields of characteristic zero. For arbitrary characteristic Okugawa4 developed necessary tools. In this paper we consider ''generic differential modules'' of type T(n), where T(n) is any fixed set of n differential operators, for an algebraic subgroup of GL(n,k) where k is a field of characteristic p. We give necessary and sufficient conditions for the existence of generic linear differential modules of type T(n). We also generalise a result of Okugawa on the decomposition of differential prime ideals in the extensions of differential polynomial rings.