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Mathematics > Number Theory

arXiv:1803.04064 (math)
[Submitted on 11 Mar 2018 (v1), last revised 8 May 2019 (this version, v2)]

Title:Class number formula for dihedral extensions

Authors:Luca Caputo, Filippo A. E. Nuccio
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Abstract: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 recovers similar formulas which have appeared in the literature as special cases. 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.
Comments: This is the version to appear in the Glasgow Mathematical Journal. All references and theorem numbering coincide with that of the published version
Subjects: Number Theory (math.NT)
MSC classes: 11R29, 11R20, 11R34
Cite as: arXiv:1803.04064 [math.NT]
  (or arXiv:1803.04064v2 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1803.04064
Journal reference: Glasgow Math. J. 62 (2020) 323-353
Related DOI: https://doi.org/10.1017/S0017089519000144

Submission history

From: Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio [view email]
[v1] Sun, 11 Mar 2018 23:21:12 UTC (28 KB)
[v2] Wed, 8 May 2019 17:25:48 UTC (30 KB)
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