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

Let p be an odd prime number and k a finite extension of image. Let K/k be a totally ramified elementary abelian Kummer extension of degree p2 with Galois group G. We determine the isomorphism class of the ring of integers in K as an image-module under some assumptions. The obtained results imply there exist extensions whose rings are image-isomorphic but not image-isomorphic, where image 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.