On the Invariant Factors of Kummer Orders in the Rings of Integers of p-adic Number Fields of Degreep2

✍ Scribed by Yoshimasa Miyata

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 356 KB
• Volume: 66
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1997.2170

✦ Synopsis

Let KÂk be an extension of degree p 2 over a p-adic number field k with the Galois group G. We study the Galois module structure of the ring O K of integers in K. We determine conditions under which the invariant factors of Kummer orders O K t in O K of two extensions coincide with each other and give two examples, one of which shows there exist Kummer extensions K and L with D(K)=D(L) such that O K and O L are not Z p G-isomorphic. The other shows the existence of extensions F and K such that O F and O K are isomorphic over Z p G but not over o k G. 1997 Academic Press

Let p be an odd prime. Let k be a p-adic number field and KÂk be a cyclic Kummer extension of degree p 2 . Then the Galois group of KÂk is isomorphic to a cyclic group G of order p 2 , so the ring O K of integers of K is an o k G-module via this isomorphism. Fro lich [4] defined the Kummer order O K t of O K as follows. Let / be a character of G and define

Then the order O K t is defined to be the direct sum

where / runs over characters of G. Let ? be a prime element of k and

Reiner [2, Section 4D]). Then

The main aim of this paper is to obtain conditions under which the invariant factors D(F ) and D(K) of two Kummer extensions F and K are the same (Theorem 12). Elder [3] obtained the decomposition of O K into Z p G-indecomposable modules. We apply his theorem with theorems obtained in this paper to article no. NT972170 314 0022-314XÂ97 25.00