Prime factors of class number of cyclotomic fields
type: Item
instance of: scholarly article
title: Prime factors of class number of cyclotomic fields
zbMATH Open document ID: 1163.11078
author: Tetsuya Taniguchi
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-04-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10848
review text: Let \(p\) be an odd prime, \(\zeta\) a primitive \(p\)-th root of unity, and \(K = \mathbb Q(\zeta)\) the cyclotomic field defined by \(\zeta\). Kummer's class number formula for the minus class number \(h^-(K)\) involves a product of values of the polynomial \(F(x) = \sum_{k=0}^{p-2} r_{-k}x^k\), where \(r_k\) is the smallest positive residue of \(r^k \bmod p\). Roland Quême asked whether the \(\ell\)-rank of the class group of \(K\) (for primes \(\ell \neq p\)) is equal to the degree of \(\gcd(F(x),x^N+1)\), where \(N = (p-1)/2\). Here the author shows that \(\ell = 3\) and \(p = 3299\) provides a counterexample: already the quadratic subfield \(\mathbb Q(\sqrt{-p}\,)\) of \(K\) has \(3\)-rank \(2\), whereas \(\gcd(F(x),x^N+1) = x+1\). The weaker questions whether the degree of \(\gcd(F(x),x^N+1)\) is a lower bound for the \(\ell\)-rank of the class group, or whether it is equal to the \(\ell\)-rank of a suitably chosen relative class group, remain open.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 5543174
zbMATH Keywords: cyclotomic field
zbMATH Keywords: minus class number
zbMATH Keywords: rank
zbMATH Keywords: class group
MaRDI profile type: Publication
OpenAlex ID: W2318644547
cites work: Resultants of Cyclotomic Polynomials
cites work: Q4214902
cites work: Prime Factors of Cyclotomic Class Numbers
cites work: Q4692759
cites work: Minus class groups of the fields of the 𝑙-th roots of unity
cites work: Q5690489
DOI: 10.5802/JTNB.639
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description: scientific article; zbMATH DE number 5543174