The positive discriminant case of Nagell's theorem for certain cubic orders
type: Item
instance of: scholarly article
title: The positive discriminant case of Nagell's theorem for certain cubic orders
zbMATH Open document ID: 1219.11163
author: M. Spindler
author: S. B. Mulay
published in: Journal of Number Theory
publication date: 2011-02-18T00:00:00Z
review text: Let \(a\) and \(b\) be two integers and \(f = X^{3} + aX^{2} + bX + 1\) be a cubic polynomial which is irreducible over the rational numbers. Let \(u\) be a real root of \(f\) and \(R:= Z[u]\) be the cubic order generated by \(u\). An element \(c\) of \(R\) is said to be a fundamental unit of \(R\), provided its absolute value \(|c|\) is a member of some basis of the torsion-free part of the unit group of \(R\). A natural question is whether \(u\) is necessary a fundamental unit of \(R\) in this sense. The case where \(f\) has a negative discriminant (case where \(u\) is the unique real root of \(f\)), was treated by \textit{T. Nagell} in his paper [``Zur Theorie der kubischen Irrationalitäten.'' Acta Math. 55, 33--65 (1930; JFM 56.0168.04)]. In this paper, the authors study the case where \(f\) has a positive discriminant (where the other two conjugates of \(u\) are also real); and they prove that \(u\) is always a fundamental unit of the order \(Z[u]\). The exceptions are shown to consist of a single infinite family together with one sporadic case. Their result is analogue of Nagell's result in the case where \(f\) has a negative discriminant (where \(u\) is the unique real root of \(f\)).
reviewed by: Abdelmalek Azizi
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 5853441
zbMATH Keywords: Cubic polynomials
zbMATH Keywords: cubic orders
zbMATH Keywords: discriminant
zbMATH Keywords: trace forms
zbMATH Keywords: fundamental units.
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.09.007
OpenAlex ID: W2075391544
cites work: Advanced Topics in Computional Number Theory
cites work: Q5838513
cites work: Fundamental units in a family of cubic fields
cites work: Systems of fundamental units in cubic orders
cites work: The class number one problem for some totally complex quartic number fields
cites work: Q3863959
DOI: 10.1016/J.JNT.2010.09.007
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description: scientific article; zbMATH DE number 5853441