A Valuation Criterion for Normal Basis Generators of Hopf-Galois Extensions in Characteristic p

Abstract

Let S/R be a finite extension of discrete valuation rings of characteristic p>0, and suppose that the corresponding extension L/K of fields of fractions is separable and is H-Galois for some K-Hopf algebra H. Let DS/R be the different of S/R. We show that if S/R is totally ramified and its degree n is a power of p, then any element ρ of L with vL(ρ)≡−vL(DS/R)−1(modn) generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.