A class number problem in the cyclotomic \(\mathbb Z_3\)-extension of \(\mathbb Q\)
type: Item
instance of: scholarly article
title: A class number problem in the cyclotomic \(\mathbb Z_3\)-extension of \(\mathbb Q\)
zbMATH Open document ID: 1205.11116
author: Takayuki Morisawa
published in: Tokyo Journal of Mathematics
publication date: 2010-07-09T00:00:00Z
review text: Let \(\Omega_n=\mathbb Q\left(2\cos\bigl(\frac{2\pi}{3^{n+1}}\bigr)\right)\). This is the \(n\)-th lyer of the cyclotomic \(\mathbb Z_3\)-extension of \(\mathbb Q\). Let \(h_n\) be the class number of \(\Omega_n\). Let \(l\geq 5\) be a prime number and \(3^s\) the exact power of \(3\) dividing \(l^2-1\). Put \[ m_l=3s+2+[\log_3(l-1)]+\left[\log_3\frac{l-1}{2}\right]+[\log_3(2s+1+[\log_3(l-1)])], \] where \([x]\) denotes the greatest integer not exceeding \(x\). The author proves that if \(l\) does not divide \(h_{m_l}\), then \(l\) does not divide \(h_n\) for any positive integer \(n\). As a corollary, if \(l<10000\) then \(l\) does not divide \(h_n\) for any positive integer \(n\).
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 5734881
zbMATH Keywords: cyclotomic \(\mathbb Z_3\)-extension of \(\mathbb Q\)
zbMATH Keywords: class number
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1264170249
OpenAlex ID: W2056851338
cites work: Weber's Class Number Problem in the Cyclotomic ℤ2-Extension of ℚ
cites work: Algorithmic Number Theory
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cites work: A note on class numbers of algebraic number fields
cites work: Class Number Computations of Real Abelian Number Fields
cites work: Q3864603
cites work: Q5690489
DOI: 10.3836/TJM/1264170249
description: scientific article; zbMATH DE number 5734881