Some explicit upper bounds on the class number and regulator of a cubic field with negative discriminant
type: Item
instance of: scholarly article
title: Some explicit upper bounds on the class number and regulator of a cubic field with negative discriminant
zbMATH Open document ID: 0649.12003
author: Pierre Barrucand
author: John H. Loxton
author: Hugh C. Williams
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pjm/1102699099
review text: Let \(K\) be a cubic number field with regulator \(R\) and discriminant \(\Delta <0\); then \(\Delta = df^2\), where \(d\) is the discriminant of a quadratic field. Let \(h\) be the number of ideal classes of the ring of integers of \(K\). The authors obtain explicit bounds on the product \(hR\), in terms of \(d\) and \(f\). For example, if \(K\) is pure cubic then, \(hR<(2f\cdot \log (f)+f\cdot \log (3))/6\), and if, in addition, \(\Delta <-6912\), then \(hR>0.44(| \Delta | /27)^{1/4}/\log (| \Delta | /27)\).
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R16
zbMATH DE Number: 4059528
zbMATH Keywords: cubic number field
zbMATH Keywords: upper bounds
zbMATH Keywords: class number
zbMATH Keywords: regulator
zbMATH Keywords: discriminant
reviewed by: Ezra Brown
MaRDI profile type: Publication
OpenAlex ID: W1984498112
DOI: 10.2140/PJM.1987.128.209
description: scientific article; zbMATH DE number 4059528