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

TL;DR

This paper establishes a valuation-based criterion for identifying normal basis generators in Hopf-Galois extensions of local fields in characteristic p, extending classical Galois theory results.

Contribution

It generalizes Elder's criterion from Galois to Hopf-Galois extensions, providing a new valuation condition for generators in totally ramified extensions of degree a power of p.

Findings

The criterion applies to totally ramified extensions of degree p^k.

Any element with valuation congruent to -v(D_{S/R})-1 mod n generates the extension as an H-module.

The result is shown to be optimal and extends previous Galois theory work.

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.