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 L such that the Galois group of L/K is the dihedral group of order 2p. We shall prove a complete characterization of the freeness of the ring of integers OL over its associated order A L/K in the unique Hopf-Galois structure on L/K, which parallels 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 OL as A L/K -module.