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

Abstract

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(π K p) for some prime element π K ∈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 𝔓 L n as a module over its associated order 𝔄 K[G] (n)={x∈K[G]:x𝔓 L n ⊆𝔓 L n } 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.