Sur la monogénéité de l'anneau des entiers de certains corps de rayon. (On the monogenicity of the ring of integers of certain ray fields)
type: Item
instance of: scholarly article
title: Sur la monogénéité de l'anneau des entiers de certains corps de rayon. (On the monogenicity of the ring of integers of certain ray fields)
zbMATH Open document ID: 0712.11063
DOI: 10.1007/BF01168377
author: Jean Cougnard
author: Vincent Fleckinger
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155387
review text: The authors show that the main conjecture of monogenicity (see the previous review) is not true in general. In particular, let \(k={\mathbb{Q}}(\sqrt{-19})\), \(\omega =(1+\sqrt{-19})/2\). The ideal 7 splits in k/\({\mathbb{Q}}\) and \(P_ 7\), the ideal generated by \(q=1+\omega\), is one of the ideals dividing it. Then the ring of integers of the extension K of k of ray \(P_ 7\) does not possess a power basis over \({\mathbb{Z}}_ k\). To prove this the authors reduce the problem to showing the nonexistence of integer points of \({\mathbb{Z}}_ k\) on an elliptic curve and use the method of Baker and Ellison [\textit{W. J. Ellison}, \textit{F. Ellison}, \textit{J. Pesek}, \textit{C. E. Stahl} and \textit{D. S. Stall}, J. Number Theory 4, 107-117 (1972; Zbl 0236.10010)] to resolve the problem. Alternatively, the authors show that they could have used the latest results of \textit{M. Mignotte} and \textit{M. Waldschmidt} [preprint, Université de Strasbourg, 1989] to obtain much smaller upper bounds for the magnitude of the solutions and reduce the calculations substantially.
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 4171049
zbMATH Keywords: Baker's method
zbMATH Keywords: Hilbert class field
zbMATH Keywords: ray class field
zbMATH Keywords: nonexistence of integer points on an elliptic curve
zbMATH Keywords: monogenicity
zbMATH Keywords: ring of integers
reviewed by: Ray Phillip Steiner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4171049