Galois scaffolding in one-dimensional elementary abelian extensions

Abstract

Abstract. 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.

Ramified extensions of degree p and their Hopf–Galois module structure

“…The definition of a scaffold, as presented in [BCE14], was still evolving when the term, Galois scaffold, was coined in [Eld09]. The intuition, as articulated in [Eld09,§1], was that extensions with Galois scaffolds are somehow extensions that are no more complicated than ramified cyclic extensions of degree p. A more mature intuition is now available and is articulated in [BCE14,§1].…”

“…The definition of a scaffold, as presented in [BCE14], was still evolving when the term, Galois scaffold, was coined in [Eld09]. The intuition, as articulated in [Eld09,§1], was that extensions with Galois scaffolds are somehow extensions that are no more complicated than ramified cyclic extensions of degree p. A more mature intuition is now available and is articulated in [BCE14,§1]. Still the first intuition is useful, and now that scaffolds have been defined more broadly than just for Galois extensions and classical Galois module theory, the question arose whether scaffolds are similarly present in ramified extensions of degree p that are not Galois and their Hopf-Galois structures.…”

Scaffolds and generalized integral Galois module structure

“…These scaffolds all have precision ∞, apart from those on cyclic extensions of degree p 2 in [BE13]. The main result of [Eld09] is the existence of a Galois scaffold for a certain class of arbitrarily large elementary abelian extensions in characteristic p (the "near onedimensional extensions"). The Galois module structure of the valuation ring in such extensions L/K is investigated in [BE14], where a necessary and sufficient condition (in terms of the b i ) is given for O L to be free over A L/K .…”

“…When the residue field of K is perfect, we know from [Eld09] that Galois scaffolds exist for all totally ramified biquadratic extensions in characteristic 2, and for all totally and weakly ramified p-extensions in characteristic p. To illustrate the sort of explicit information our methods can yield, we examine these two classes of extensions in detail (see Theorems 4.1 and 4.2). However, in this paper we are not primarily concerned with the problem of actually constructing A-scaffolds.…”

“…This enables us to give an explicit construction for a class of extensions in characteristic 0 which admit Galois scaffolds. These have elementary abelian Galois groups of arbitrarily large rank, and are the analogs in characteristic 0 of the near one-dimensional extensions in characteristic p constructed in [Eld09]. They include the totally ramified biquadratic extensions and the totally and weakly ramified p-extensions satisfying some additional hypotheses.…”