Realizable Classes of Quaternion Extensions of Degree 4ℓ

✍ Scribed by Bouchaı̈b Sodaı̈gui

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 180 KB
• Volume: 80
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2448

✦ Synopsis

Let k be a number field and O k its ring of integers. Let 1 be the generalized quaternion group of order 4l, where l is an odd prime number. Let M be a maxi-

and Cl(M) its class group. We denote by R(M) the subset of Cl(M) formed by the realizable classes in the sense of McCulloh. In this article we define two subsets of R(M) and show that they are subgroups of Cl(M) provided that 2 and l are unramified in kÂQ. 2000 Academic Press 1. INTRODUCTION

We continue to study the set of Galois module classes of rings of integers in Galois extensions of algebraic number fields. For previous work in this direction see [6,10,12,13].

For every number field E, O E denotes its ring of algebraic integers.

Let k be a number field and 1 a finite group. Let M be a maximal

and Cl(M) its class group. We denote by R(M) the set of realizable classes, that is the set of classes c # Cl(M) such that there exists a Galois extension NÂk at most tamely ramified (i.e., tame), with Galois group isomorphic to 1 and the class of

In this article, we attempt to determine the structure of R(M) in the case that 1=H 4l the generalized quaternion group of order 4l, where l is an odd prime number, defined by the presentation H 4l =(_, {: _ 2l =1, { 2 =_ l , {_{ &l =_ &1 ) (For this terminology see [1], (1.24) Examples, (ii), p. 13). We do not succeed in determining completely that structure, but we will define below two subsets of R(M) and show that they are subgroups of Cl%(M) provided that 2 and l are unramified in kÂQ.