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.