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

✍ Scribed by Yoshimasa Miyata

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 212 KB
• Volume: 125
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.12.005

✦ Synopsis

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 oG-module under some assumptions. The obtained results imply there exist extensions whose rings are Z p G-isomorphic but not oG-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.