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

Cyclic, ramified extensions L/K of degree p of local fields with residue characteristic p are fairly well understood. They are defined by an Artin-Schreier equation, unless char(K) = 0 and L = K((p)root pi(K)) for some prime element pi(K) is an element of K. Moreover, through the work of Bertrandias-Ferton (char(K) = 0) and Aiba (char(K) = p), much is known about the Galois module structure of the ideals in such extensions: the structure of each ideal PLn as a module over its associated order K-A[G] (n) = {x is an element of K[G] : xPL(n) subset of PLn} where G = Gal(L/K). The purpose of this paper is to extend these results to separable, ramified extensions of degree p that are not Galois.