Ramified extensions of degree $p$ and their {H}opf-{G}alois module structure

Cyclic, ramified extensions $L/K$ of degree $p$ of local fields with residue characteristic $p$ are fairly well understood. Unless $\mbox{char}(K)=0$ and $L=K(\sqrt[p]{\pi_K})$ for some prime element $\pi_K\in K$, they are defined by an Artin-Schreier equation. Additionally, through the work of Ferton, Aiba, de Smit and Thomas, and others, much is known about their Galois module structure of ideals, the structure of each ideal $\mathfrak{P}_L^n$ as a module over its associated order $\mathfrak{A}_{K[G]}(n)=\{x\in K[G]:x\mathfrak{P}_L^n\subseteq \mathfrak{P}_L^n\}$ where $G=\mbox{Gal}(L/K)$. This paper extends these results to separable, ramified extensions of degree $p$ that are not Galois.