On Vandiver's conjecture
type: Item
instance of: scholarly article
title: On Vandiver's conjecture
zbMATH Open document ID: 0828.11059
DOI: 10.1007/BF02940778
author: Stanislav Jakubec
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1996-01-07T00:00:00Z
review text: The main theorem of the paper is the following: Let \(p\) be a prime of the form \(n^4 + 5n^3 + 15n^2 + 25n + 25\) and let \(K\) be the quintic subfield of \(\mathbb{Q} (\zeta_p + \zeta_p^{- 1})\). There exists \(N_0\) such that for \(p > N_0\), \(p\) divides the class number \(h_K \) if and only if \(p\) divides a Bernoulli number \(B_j\) with \(j = (p - 1)/5\), \(2(p - 1)\), \(3(p - 1)/5\), \(4(p - 1)/5\). This family of quintic fields was discovered by \textit{E. Lehmer} [Math. Comput. 50, No. 182, 535-541 (1988; Zbl 0652.12004)]. \textit{R. Schoof} and the reviewer [Math. Comput. 50, No. 182, 543-556 (1988; Zbl 0649.12007)] determined a set of fundamental units of \(K\) and gave examples where \(h_K\) has a prime factor larger than \(p\). The interesting aspect of the present theorem is that the divisibility of a Bernoulli number by \(p\) implies that the class number of the real field \(K\) is divisible by \(p\). Usually such criteria apply only to relative class numbers of imaginary fields. The proof constructs a nontrivial homomorphism from the units of \(K\) to \(\mathbb{Z}/p \mathbb{Z}\) that vanishes on the cyclotomic units when \(p\) divides the appropriate Bernoulli number. This technique is also used to study Vandiver's conjecture under various hypotheses.
reviewed by: Lawrence C. Washington
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 706666
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: Bernoulli number
zbMATH Keywords: quintic fields
zbMATH Keywords: class number
zbMATH Keywords: Vandiver's conjecture
Wikidata QID: Q123290532
MaRDI profile type: Publication
cites work: Q5833279
cites work: Q3234208
cites work: Q5798085
cites work: The congruence for Gauss period
cites work: Gaussian Periods and Units in Certain Cyclic Fields
cites work: Connection Between Gaussian Periods and Cyclic Units
cites work: A class number congruence for cyclotomic fields and their subfields
cites work: Nombre de classes d'une extension cyclique reelle deQ, de degre 4 ou 6 et de conducteur premier
cites work: Majorization of the number of classes of a cyclic cubic field of prime conductor
cites work: Quintic Polynomials and Real Cyclotomic Fields with Large Class Number
cites work: Q3943047
description: scientific article; zbMATH DE number 706666