G. Griffith Elder

Galois scaffolding in one-dimensional elementary abelian extensions

A Galois scaffold is defined to be a variant of a normal basis that allows for an easy determination of valuation and thus has implications for the questions of the Galois module structure. We introduce a class of elementary abelian p-extensions of local function fields of characteristic p, which we call one-dimensional and which should be considered no more complicated than cyclic degree p extensions, and show that they, just as cyclic degree p extensions, possess a Galois scaffold.

Integral Galois module structure for elementary abelian extensions with a Galois scaffold

This paper justifies an assertion in [Eld09] that Galois scaffolds make the questions of Galois module structure tractable. Let k be a perfect field of characteristic p and let K = k((T )). For the class of characteristic p elementary abelian p-extensions L/K with Galois scaffolds described in [Eld09], we give a necessary and sufficient condition for the valuation ring OL to be free over its associated order AL/K in K[Gal(L/K)]. Interestingly, this condition agrees with the condition found by Y. Miyata, concerning a class of cyclic Kummer extensions in characteristic zero.

Sufficient conditions for large Galois scaffolds

Abstract Let L / K be a finite, Galois, totally ramified p-extension of complete local fields with perfect residue fields of characteristic p > 0 . In this paper, we give conditions, valid for any Galois p-group G = Gal ( L / K ) (abelian or not) and for K of either possible characteristic (0 or p), that are sufficient for the existence of a Galois scaffold. The existence of a Galois scaffold makes it possible to address questions of integral Galois module structure, which is done in a separate paper [BCE] . But since our conditions can be difficult to check, we specialize to elementary abelian extensions and extend the main result of [Eld09] from characteristic p to characteristic 0. This result is then applied, using a result of Bondarko, to the construction of new Hopf orders over the valuation ring O K that lie in K [ G ] for G an elementary abelian p-group.