Weber's class number problem in the cyclotomic \(\mathbb Z_2\)-extension of \(\mathbb Q\). II.
type: Item
instance of: scholarly article
title: Weber's class number problem in the cyclotomic \(\mathbb Z_2\)-extension of \(\mathbb Q\). II.
zbMATH Open document ID: 1223.11133
author: Takashi Fukuda
author: Keiichi Komatsu
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2011-03-07T00:00:00Z
full work available at URL: https://eudml.org/doc/116408
review text: Let \(\zeta_n=\exp\left(\frac{2\pi i}{2^n}\right) \), \({\mathbb Q}_n= {\mathbb Q}(\zeta_{n+2}+\zeta_{n+2}^{-1})\), \(h_n\) the class number of \({\mathbb Q}_n\). Let \(l\) be an odd prime number, \(2^c\) the exact power of \(2\) dividing \(l-1\) or \(l^2-1\) according as \(l\equiv 1\pmod 4\) or not, \(m_l=2c-3+[\log_2l] \). The authors prove that \(l\) does not divide \(\frac{h_n}{h_{m_l}} \) for any integer \(n\geq m_l\). If \(l\equiv \pm 9\pmod{16}\), then \(l\) does not divide \(h_n\) for all \(n\geq 1\). For Part I, see Exp. Math. 18, No. 2, 213--222 (2009; Zbl 1189.11033), Part III, Int. J. Number Theory 7, No. 6, 1627--1635 (2011; Zbl 1226.11119).
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 5862104
zbMATH Keywords: cyclotomic \(\mathbb Z_2\)-extension of \(\mathbb Q\)
zbMATH Keywords: class number
MaRDI profile type: Publication
OpenAlex ID: W2461455999
cites work: Numerische Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper
cites work: A numerical study of Weber's real class number calculation
cites work: Weber's Class Number Problem in the Cyclotomic ℤ2-Extension of ℚ
cites work: Ideal Class Groups of Iwasawa-Theoretical Abelian Extensions Over the Rational Field
cites work: The ideal class group of the basic \(\mathbb Z_p\)-extension over an imaginary quadratic field
cites work: Triviality in ideal class groups of Iwasawa-theoretical abelian number fields
cites work: Primary components of the ideal class group of an Iwasawa-theoretical Abelian number field
cites work: Certain primary components of the ideal class group of the \(\mathbb Z_p\)-extension over the rationals
cites work: The ideal class group of the \(\mathbb Z_p\)-extension over the rationals
cites work: Q3864603
cites work: Class Number Computations of Real Abelian Number Fields
cites work: Class numbers and \(\mathbb Z_p\)-extensions
cites work: The non-p-part of the class number in a cyclotomic \(\mathbb{Z}_p\)-extension
DOI: 10.5802/JTNB.720
description: scientific article; zbMATH DE number 5862104