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.

Abstract. We explicitly describe, in terms of indecomposable Z 2 [G]-modules, the Galois module structure of ideals in totally ramified biquadratic extensions of local number fields with only one break in their ramification filtration. This paper completes work begun in [Elder: Canad.

Let LÛK be a finite Galois extension of local fields which are finite extensions of Qp, the field of p-adic numbers. Let Gal(LÛK) = G, and Ó L and Zp be the rings of integers in L and Qp, respectively. And let Ÿ L denote the maximal ideal of Ó L . We determine, explicitly in terms of specific indecomposable Zp[G]-modules, the Zp[G]-module structure of Ó L and Ÿ L , for L, a composite of two arithmetically disjoint, ramified cyclic extensions of K, one of which is only weakly ramified in the sense of Erez [6].

Let L/K be a finite, totally ramified p-extension of complete local fields with residue fields of characteristic p > 0, and let A be a K-algebra acting on L. We define the concept of an A-scaffold on L, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where L/K was Galois and A = K[G] for G = Gal(L/K). When a suitable A-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, for a given fractional ideal to be free over its associated order in A. We also show how to determine the number of generators required when it is not free, along with the embedding dimension of the associated order. In the Galois case, the numerical parameters are the ramification breaks associated with L/K. We apply these results to biquadratic Galois extensions in characteristic 2, and to totally and weakly ramified Galois p-extensions in characteristic p. We also apply our results to the non-classical situation where L/K is a finite primitive purely inseparable extension of arbitrary exponent that is acted on, via a higher derivation (but in many different ways), by the divided power K-Hopf algebra.

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.