On Galois structure of the integers in elementary abelian extensions of local number fields

Abstract

Let p be an odd prime number and k a finite extension of Q p . Let K / k be a totally ramified elementary abelian Kummer extension of degree p 2 with Galois group G. We determine the isomorphism class of the ring of integers in K as an o G -module under some assumptions. The obtained results imply there exist extensions whose rings are Z p G -isomorphic but not o G -isomorphic, where Z p is the ring of p-adic integers. Moreover we obtain conditions that the rings of integers are free over the associated orders and give extensions whose rings are not free.