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CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  21 May 2019

LUCA CAPUTO  and
FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO Open the ORCID record for FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO [Opens in a new window]
FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO
Affiliation:
Univ Lyon, Université Jean Monnet Saint-Étienne, CNRS UMR 5208, Institut Camille Jordan, F-42023 Saint-Étienne, France e-mails: luca.caputo@gmx.com; filippo.nuccio@univ-st-etienne.fr
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Abstract

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We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

MSC classification

Primary: 11R29: Class numbers, class groups, discriminants 11R20: Other abelian and metabelian extensions
Secondary: 11R34: Galois cohomology 11R37: Class field theory

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 2 , May 2020 , pp. 323 - 353
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

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