Class numbers of real cyclotomic fields of composite conductor
type: Item
instance of: scholarly article
title: Class numbers of real cyclotomic fields of composite conductor
zbMATH Open document ID: 1300.11109
DOI: 10.1112/S1461157014000382
author: John C. Miller
published in: LMS Journal of Computation and Mathematics
publication date: 2014-09-05T00:00:00Z
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R80
Mathematics Subject Classification ID: 11Y40
zbMATH DE Number: 6340508
zbMATH Keywords: class number
zbMATH Keywords: maximal totally real subfield of cyclotomic fields
zbMATH Keywords: root discriminant
zbMATH Keywords: conductor
zbMATH Keywords: Hilbert class field
MaRDI profile type: Publication
cites work: Class numbers of real cyclotomic fields of prime conductor
cites work: Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions : a survey of recent results
full work available at URL: s1461157014000382
OpenAlex ID: W2332358942
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review text: Let \(K\) denote a number field of degree n over \(\mathbb Q\). Let \(d(K)\) denote its discriminant. The \textit{root discriminant} \(\text{rd}(K)\) of \(K\) is defined by \(\text{rd}(K) := |d(K)|^{1/n}\). Let \(\varphi\) be the Euler phi-function. Exploiting Odlyzko's discriminant lower bounds, \textit{J. M. Masley} [Compos. Math. 37, 297--319 (1978; Zbl 0428.12003)] and \textit{F. J. van der Linden} [Math. Comput. 39, 693--707 (1982; Zbl 0505.12010)] established the class numbers of all real cyclotomic fields of composite conductor \(m\), provided that \(m \leq 200\), \(\varphi(m) \leq 72\) and \(m\neq 148, 152\). For fields of larger degree or conductor, the root discriminant becomes too large for their methods. To overcome this problem, the author establishes by an analytic approach a lower bound on some sums over prime ideals of the Hilbert class fields, which in turn allows to obtain an upper bound on the class number. NEWLINENEWLINENEWLINE The main result is:NEWLINENEWLINENEWLINE Theorem. Let \(m\) be a composite integer, \(m\not\equiv 2\bmod 4\), and let \(\mathbb Q(\zeta_m)^+\) the maximal real subfield of the \(m\)th cyclotomic field \(\mathbb Q(\zeta_m)\). Then the class number \(h^+\) of \(\mathbb Q(\zeta_m)^+\) is \(1\) if \(\varphi(m)\leq 116\) and \(m\not\in\{ 136,145,212\}\), \(2\) if \(m=136\), \(2\) if \(m=145\) and \(1\) if \(m=256\). Furthermore, under the generalized Riemann hypothesis (GRH), \(h^+_{212} = 5\) and \(h^+_{512} = 1\).NEWLINENEWLINEAs a consequence, the real cyclotomic field of conductor \(420\) has class number \(1\), which is the largest conductor for which the class number of a cyclotomic field has been calculated unconditionally. Note that this theorem on composite conductors complements the earlier results of the author on real cyclotomic fields of prime conductor [``Real cyclotomic fields of prime conductor and their class numbers'', Math. Comput. (to appear), \url{arXiv:1407.2373}].NEWLINENEWLINEThe result of the author in …
reviewed by: Roland Quême
description: scientific article; zbMATH DE number 6340508