On some cubic or quartic algebraic units
type: Item
instance of: scholarly article
title: On some cubic or quartic algebraic units
zbMATH Open document ID: 1211.11121
published in: Journal of Number Theory
publication date: 2010-04-08T00:00:00Z
review text: Let \(\varepsilon\) be an algebraic unit that is not a complex root of unity. Assume that the rank of the unit group of the order \(\mathbb Z[\varepsilon]\) is equal to one. Then \(\mathbb Q(\varepsilon)\) is either (i) a real quadratic number field, (ii) a not totally real cubic number field, or (iii) a totally imaginary quartic number field. The subject of the paper is to answer the question: is \(\varepsilon\) a fundamental unit of \(\mathbb Z[\varepsilon]\)? The answer for the case (ii) is in [\textit{T. Nagell}, Acta Math. 55, 33--65 (1930; JFM 56.0168.04)]; a new proof of it is given in [\textit{S. Louboutin}, J. Number Theory 121, No. 1, 30--39 (2006; Zbl 1171.11057)]. In [\textit{J. Ramanujan}, J. Ramanujan Math. Soc. 23, No. 2, 191--210 (2008; Zbl 1165.11076)] the present author gave the answer for the case (i). In [\textit{S. M. Park} and \textit{G. N. Lee}, J. Number Theory 129, No. 6, 1338--1349 (2009; Zbl 1167.11040)] the answer for (iii) is given. In the paper under review, there is no new result, but the author gives a simpler proof of the key result for the cases (ii) and (iii).
reviewed by: Bouchaïb Sodaïgui
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R09
zbMATH DE Number: 5690731
zbMATH Keywords: unit
zbMATH Keywords: polynomial
zbMATH Keywords: discriminant
zbMATH Keywords: cubic number field
zbMATH Keywords: quartic number field
author: Stéphane Louboutin
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.09.002
OpenAlex ID: W1972228998
cites work: The class-number one problem for some real cubic number fields with negative discriminants
cites work: Q3597606
cites work: The class number one problem for some totally complex quartic number fields
cites work: Q5690489
DOI: 10.1016/J.JNT.2009.09.002
description: scientific article; zbMATH DE number 5690731