On Brauer-Kuroda type relations of \(S\)-class numbers in dihedral extensions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1270.11115
DOI: 10.1515/CRELLE.2011.152
author: Alex Bartel
publication date: 2012-08-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/0904.2416
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 19F27
zbMATH DE Number: 6063241
zbMATH Keywords: class number
zbMATH Keywords: S-class group
zbMATH Keywords: dihedral extension
zbMATH Keywords: Dokchitser constants
MaRDI profile type: Publication
arXiv ID: 0904.2416
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cites work: Brauer-Kuroda relations for \(S\)-class numbers
title: On Brauer-Kuroda type relations of \(S\)-class numbers in dihedral extensions
published in: Journal für die Reine und Angewandte Mathematik
review text: Let \(F/k\) be a Galois extension of number fields, and assume that its Galois group is the dihedral group \(D_{2q}\) of order \(2q\), where \(q\) is an odd integer. Let \(S\) be a finite set of primes in \(F\), and assume that \(S\) is fixed by the action of the Galois group and that it contains all the archimedean primes. Let \({\mathcal O} = {\mathcal O}_S\) denote the ring of \(S\)-integers, i.e., the ring of all algebraic numbers in \(F\) which are integral at all places not in \(S\), let \(C_S(F)\) denote the \(S\)-class group of \(F\), which is, by definition, the class group of the Dedekind ring \({\mathcal O}\) (reviewer's remark: this definition excludes the \(S\)-class groups for sets \(S\) not necessarily containing all Archimedean primes, in particular, it excludes the class groups in the struct sense), and let \(h_S(F) = \# C_S(F)\) denote the \(S\)-class number.NEWLINENEWLINEThe author's main result is the following generalization of a class number formula for \(D_{2q}\)-extensions obtained by \textit{L. Caputo} and \textit{F. Nuccio} [``On fake \(\mathbb Z_p\) extensions of number fields'', \url{arXiv:0807.1135}]: Let \(p\) be an odd prime number, \(F/k\) be a \(D_{2p}\)-extension of number fields, let \(K/K\) be the quadratic subextension inside \(F/k\), and let \(L\) and \(L'\) be distinct intermediate extensions of degree \(p\) over \(k\). As above, let \({\mathcal O}_F\) denote the ring of \(S\)-integers in \(F\), and \({\mathcal O}_F^\times\) its unit group. Let \(r_s\) denote the rank of the group of \(S\)-units, and let \(a(F/K,S)\) denote the number of primes of \(k\) that lie below those in \(S\) and have decomposition group \(D_{2p}\). Finally, set \(\delta = 3\) or \(1\) according as \(L/K\) is generated by a \(p\)-th root of a non-torsion \(S\)-unit or not. Then NEWLINE\[NEWLINE \frac{h_S(F) h_S(k)^2}{h_S(K) h_S(L)} = p^{\alpha/2} \cdot [{\mathcal O}_F^\times : {\mathcal O}_L^\times{\mathcal O}_{L'}^\times {\mathcal O}_K^\times], NEWLINE\…
reviewed by: Franz Lemmermeyer
description: scientific article; zbMATH DE number 6063241