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Cyclic, ramified extensions $L/K$ of degree $p$ of local fields with residuecharacteristic $p$ 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 byan Artin-Schreier equation. Additionally, through the work of Ferton, Aiba, deSmit and Thomas, and others, much is known about their Galois module structureof ideals, the structure of each ideal $\mathfrak{P}_L^n$ as a module over itsassociated 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 resultsto separable, ramified extensions of degree $p$ that are not Galois.

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