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

Abstract

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\tilde{L}$ such that the Galois group of $\tilde{L}/K$ 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.