On the 2-part of the ideal class group of the cyclotomic \(\mathbb Z_p\)-extension over the rationals
type: Item
instance of: scholarly article
title: On the 2-part of the ideal class group of the cyclotomic \(\mathbb Z_p\)-extension over the rationals
zbMATH Open document ID: 1222.11126
author: Humio Ichimura
author: Shoichi Nakajima
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2010-11-30T00:00:00Z
review text: Let \(p\not=l\) be two primes with \(p>2\). For a number field \(F\), let \(F_{\infty}/F\) the cyclotomic \(\mathbb{Z}_p\)-extension and \(F_n\) the \(n\)-th layer of \(F_{\infty}/F\). Moreover, we denote by \(h_{F_n}\) the class number of \(F_n\), and by \(Cl_{F_{\infty}}\) the direct limit of the ideal class groups of \(F_n\). It is a classical result of \textit{L. C. Washington} [Invent. Math. 49, 87--97 (1978; Zbl 0403.12007)] that the quotient \(h_{F_n} / h_{F_{n-1}}\) is not divisible by \(l\) for \(n > N(p)\) for sufficiently large \(N(p)\), i.e. the \(l\)-part \(Cl_{F_{\infty}}(l)\) of \(Cl_{F_{\infty}}\) is a finite abelian group. In the paper under review the authors show that \(Cl_{F_{\infty}}(2)\) is trivial if \(p < 500\) and either \(F = \mathbb{Q}\) or the order of \(2 \bmod p\) is even and \(F\) is the maximal subfield of the \(p\)-th cyclotomic field \(\mathbb{Q}(\zeta_p)\) such that \([F:\mathbb{Q}]\) is a power of two. For this, they use an expicit bound \(N(p)\) as above which is easily obtained from work of \textit{K. Horie} [J. Lond. Math. Soc., II. Ser. 66, No. 2, 257--275 (2002; Zbl 1011.11072); Tôhoku Math. J. (2) 61, No. 4, 551--570 (2009; Zbl 1238.11101)].
reviewed by: Andreas Nickel
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 5821521
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: class groups
zbMATH Keywords: class numbers
MaRDI profile type: Publication
full work available at URL: s12188-010-0036-x
OpenAlex ID: W2025215331
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DOI: 10.1007/S12188-010-0036-X
description: scientific article; zbMATH DE number 5821521