Computation of the first factor of the class number of cyclotomic fields
type: Item
instance of: scholarly article
title: Computation of the first factor of the class number of cyclotomic fields
zbMATH Open document ID: 0762.11039
DOI: 10.1016/0022-314X(92)90095-7
author: Gilbert Fung
author: Andrew Granville
author: Hugh C. Williams
published in: Journal of Number Theory
publication date: 1993-01-17T00:00:00Z
review text: In this interesting paper, the authors introduce a new method for computing the first factor \(h_ 1(p)\) of the class number of the \(p\)th cyclotomic field \((p\) an odd prime). As is well known, \(h_ 1(p)\) grows very fast, with about the same speed as the number \(G(p)=2p(p/4\pi^ 2)^{(p-1)/4}\). Following previous computations by \textit{D. H. Lehmer} and \textit{J. M. Masley} [Math. Comput. 32, 577--582 (1978; Zbl 0382.12003)], the authors first factorize \(h_ 1(p)\) into ``relative class numbers \(h_ e\) of degree \(e\)'', where \(e\) divides \(p-1\). When computing \(h_ e\) they use a trick which considerably shortens the procedure. They have computed \(h_ 1(p)\) in the range \(100<p<3000\) and, moreover, decomposed \(h_ 1(p)\) into factors which are often primes. The paper includes several tables extracted from the results (in full deposited in the UMT file) and there is a discussion concerning such topics as the divisibility by 2-powers of \(h_ 1(p)\) and the classical conjectures related to \(h_ 1(p)\). To the extensive bibliography given by the authors, one could add the papers by \textit{K. Iwasawa} [Nagoya Math. J. 27, 239--247 (1966; Zbl 0139.28104)] and the reviewer [Ann. Univ. Turku., Ser. A I 149, 1--8 (1971; Zbl 0215.07204)] both of which contain explicit information about the 2- divisibility of \(h_ 1(p)\). Note also the recent progress in the numerical checking of some of the conjectures mentioned; see the paper by \textit{J. P. Buhler}, \textit{R. E. Crandall}, \textit{R. Ernvall} and \textit{T. Metsänkylä} [Math. Comput. 61, No. 203, 151--153 (1993; Zbl 0789.11020)]. The authors also sketch arguments giving reasons to doubt the validity of Kummer's conjecture that \(h_ 1(p)\) be asymptotically equal to \(G(p)\). These arguments are developed further in a paper by the second author [Invent. Math. 100, No. 2, 321--338 (1990; Zbl 0701.11051)]. Moreover the authors show how an extension of recent work by D. M. Goldfeld, B. Gross and D. Zagier might …
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11Y40
zbMATH DE Number: 98393
zbMATH Keywords: relative class numbers
zbMATH Keywords: computation
zbMATH Keywords: first factor of class number
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: tables
zbMATH Keywords: divisibility by 2-powers
zbMATH Keywords: bibliography
zbMATH Keywords: Kummer's conjecture
MaRDI profile type: Publication
cites work: The Class Number of the Cyclotomic Field
cites work: On the class-number of the maximal real subfield of a cyclotomic field.
cites work: Notes on elliptic curves. II.
cites work: Maillet’s determinant
cites work: Q3900124
cites work: Q5648429
cites work: The Prime Factors of Wendt's Binomial Circulant Determinant
cites work: Q4113910
cites work: Gauss’ class number problem for imaginary quadratic fields
cites work: On the size of the first factor of the class number of a cyclotomic field
cites work: Q3324081
cites work: Q5843509
cites work: Prime Factors of Cyclotomic Class Numbers
cites work: Table of the cyclotomic class numbers ℎ*(𝑝) and their factors for 200<𝑝<521
cites work: Q4046140
cites work: Q4101931
cites work: Calculation of the First Factor of the Class Number of the Cyclotomic Field
cites work: The large sieve
cites work: A Table of the First Factor for Prime Cyclotomic Fields
cites work: Computations on the growth of the first factor for prime cyclotomic fields
cites work: Q4046106
cites work: The Calculation of a Large Cubic Class Number with an Application to Real Cyclotomic Fields
cites work: New Congruences for the Bernoulli Numbers
cites work: Class numbers of imaginary abelian number fields. III
cites work: Q3943047
cites work: The Irregular Primes to 125000
description: scientific article; zbMATH DE number 98393