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Journal of the London Mathematical Society Advance Access originally published online on September 29, 2007
Journal of the London Mathematical Society 2007 76(2):331-344; doi:10.1112/jlms/jdm053
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© 2007 London Mathematical Society

Classes de Steinitz d'extensions quaternioniennes généralisées de degré 4pr

James E. Carter

Department of Mathematics
College of Charleston
66 George Street
Charleston
SC 29424-0001
USA
carterj{at}cofc.edu

Bouchaïb Sodaïgui

Département de Mathématiques
Université de Valenciennes
Le Mont Houy
59313 Valenciennes Cedex 9
France

Let p be an odd prime number, r a natural number, and Formula the generalized quaternion group of order 4pr. Let k be a number field and Cl(k) its class group. Let Formula be the subset of Cl(k) consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of k with Galois group isomorphic to Formula. In this article, we determine Formula and show that it is a subgroup of Cl(k). In particular, Formula is the full group Cl(k) if p does not divide the class number of k, or if either k contains a primitive prth root of unity or p is unramified in k.

2000 Mathematics Subject Classification 11R33

Received June 28, 2006; published online September 29, 2007.


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