On Brauer–Kuroda type relations of S-class numbers in dihedral extensions

Bartel, A. (2012) On Brauer–Kuroda type relations of S-class numbers in dihedral extensions. Journal für die reine und angewandte Mathematik (Crelles Journal), 2012(668), pp. 211-244. (doi: 10.1515/CRELLE.2011.152)

Abstract

Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer–Kuroda relations, as a unit index. Our formula is valid for arbitrary extensions with Galois group D2q and for arbitrary Galois-stable sets of primes S, containing the Archimedean ones. Our results have curious applications to determining the Galois module structure of the units modulo the roots of unity of a D2q-extension from class numbers and S-class numbers. The techniques we use are mainly representation theoretic and we consider the representation theoretic results we obtain to be of independent interest.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bartel, Professor Alex
• Authors: Bartel, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal für die reine und angewandte Mathematik (Crelles Journal)
• Publisher: De Gruyter
• ISSN: 0075-4102
• ISSN (Online): 0075-4102
• Published Online: 05 October 2011
• Copyright Holders: Copyright © 2012 Walter de Gruyter
• First Published: First published in Journal für die reine und angewandte Mathematik (Crelles Journal) 2012(668):211-244
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher