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 D_{S/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 $\rho$ of L with $v_L(\rho )$ congruent to $-v_L(D_{S/R})-1$ mod n generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. Elder for Galois extensions.

arXiv comment11 pages; LaTeX. To appear in Journal de Theorie des Nombres de Bordeaux (special volume, Proceedings of Journees Arithmetiques, St Etienne, July 2009). The example in Section 5 has been expanded following a suggestion of the referee. Several minor clarifications have been made to the text

References in corpus (1)

• A valuation criterion for normal basis generators in local fields of characteristic $p$