Hopf Galois module structure of dihedral degree $2p$ extensions of ${\mathbb{Q}}_p$

Abstract

Let $p$ be an odd prime. For field extensions $L/{\mathbb{Q}}_p$ with Galois group isomorphic to the dihedral group $D_{2p}$ of order $2p$, we consider the problem of computing a basis of the associated order in each Hopf Galois structure and the module structure of the ring of integers ${\mathcal{O}}_L$. We solve the case in which $L/{\mathbb{Q}}_p$ is not totally ramified and present a practical method which provides a complete answer for the cases $p=3$ and $p=5$. We see that within this family of dihedral extensions, the ring of integers is always free over the associated orders in the different Hopf Galois structures.

The paper has been updated with a slightly different structure. We have fixed some mistakes and obtained different answers to the same questions