The ring of integers of Hopf-Galois degree p extensions of p-adic fields with dihedral normal closure

TL;DR

This paper characterizes when the ring of integers in certain degree p extensions of p-adic fields with dihedral Galois group is free over its associated Hopf-Galois order, extending known results from cyclic cases.

Contribution

It provides a complete characterization of the freeness of the ring of integers over the associated order in dihedral Galois extensions, generalizing cyclic case results.

Findings

Complete criteria for freeness of $\\mathcal{O}_L$ over its associated order.

Positive and negative results on module freeness.

Extension of cyclic case results to dihedral Galois groups.

Abstract

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete characterization of the freeness of the ring of integers $\mathcal{O}_L$ over its associated order $\mathfrak{A}_{L/K}$ in the unique Hopf-Galois structure on $L/K$, which is analogous to the one already known for cyclic degree $p$ extensions of $p$-adic fields. We shall derive positive and negative results on criteria for the freeness of $\mathcal{O}_L$ as $\mathfrak{A}_{L/K}$-module.