A note on class numbers of the simplest cubic fields
type: Item
instance of: scholarly article
title: A note on class numbers of the simplest cubic fields
zbMATH Open document ID: 0879.11063
author: Dongho Byeon
published in: Journal of Number Theory
publication date: 1997-07-24T00:00:00Z
review text: A simplest cubic field is defined by the irreducible polynomial over the rational number field \(\mathbb{Q}\) of the form \(f(x)= x^3+ mx^2 -(m+3)x +1\) with a rational integer \(m\). For any given rational integer \(n\geq 2\), \textit{K. Uchida} [J. Math. Soc. Japan 26, 447-453 (1974; Zbl 0281.12007)] and \textit{L. C. Washington} [Math. Comput. 48, 371-384 (1987; Zbl 0613.12002)] constructed simplest cubic fields with class numbers divisible by \(n\) under the assumption of \(3 \nmid m\). In this paper, the author intends to weaken the assumption on \(m\), and for \(m \equiv 3\), 21 mod 27, \((m>3)\) he constructs simplest cubic fields with class numbers divisible by \(n\) in a similar way. Namely, let \(a>1\) be an integer satisfying \((a,6) =1\), and \(m>3\) be an integer such that \(m \equiv 3\) or 21 (mod 27) and \(2m +3= 9a^n\). Then, he proves that the class number of the simplest cubic field defined by the irreducible polynomial \(f(x)= x^3+ mx^2- (m+3) x+1\) with such \(m\) is divisible by \(3n\).
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R16
zbMATH DE Number: 1037300
zbMATH Keywords: simplest cubic field
zbMATH Keywords: class numbers
reviewed by: Hideo Yokoi
MaRDI profile type: Publication
OpenAlex ID: W1980828683
cites work: Fundamental units in a family of cyclic cubic fields
cites work: Class numbers of cubic cyclic fields
cites work: Class Numbers of the Simplest Cubic Fields
DOI: 10.1006/JNTH.1997.2144
description: scientific article; zbMATH DE number 1037300