Hopf-Galois module structure of degree p extensions of p-adic fields

TL;DR

This paper characterizes when the ring of integers in degree p extensions of p-adic fields is free over its associated order in the Hopf-Galois structure, providing a complete criterion.

Contribution

It offers a complete characterization of the conditions for the ring of integers to be free over the associated order in Hopf-Galois structures for degree p extensions.

Findings

Provides a full criterion for freeness of the ring of integers

Characterizes Hopf-Galois module structure in degree p extensions

Enhances understanding of algebraic structures in p-adic field extensions

Abstract

Let $p$ be an odd prime number. For a degree $p$ extension of $p$-adic fields $L/K$, we give a complete characterization of the condition for the ring of integers $\mathcal{O}_L$ to be free as a module over its associated order in the unique Hopf-Galois structure on $L/K$.