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 ρ of L with vL(ρ) ≡ −vL(D S/R ) − 1 (mod n) generates L as an Hmodule. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

Canonical Nonclassical Hopf–Galois Module Structure of Nonabelian Galois Extensions

“…Now we know that it makes sense to let h i act on x ∈ L according to equation (2), and so we can prove (b):…”

“…If H is a Hopf algebra giving a Hopf-Galois structure on L/K then, as noted in the introduction, L is a free H-module of rank one. However, to our knowledge the only results comparing explicit generators of L as a module over the various Hopf algebras giving Hopf-Galois structures on L/K are those appearing in [2], which are concerned with the valuation criterion for normal basis generators in characteristic p. We shall prove the following theorem:…”

Scaffolds and integral Hopf Galois module structure on purely inseparable extensions

“…A simple example of a Galois scaffold arises when n = 1 and the break number b is relatively prime to p; in this case, if G = σ then θ 1 = σ − 1, v = b is an example of a Galois scaffold. Such scaffolds do not always exist -in fact, integer certificates may not exist, for example if L/K is unramified and π p = 1 [Byo11]. This notion of scaffold was refined in [BE13], and then again in [BCE14], the latter version being the most useful for describing the integral Galois module structure.…”

On the Galois module structure of extensions of local fields

“…Write d for the valuation of the different of the extension. Then each element x ∈ L with valuation congruent to −d − 1 modulo [L : K] is a normal basis generator for L/K.Nigel Byott has reinterpreted this result in terms of Hopf-Galois structures[35].…”