Monogenic cubic cyclic extension of the ring of integers of a quadratic field
type: Item
instance of: scholarly article
title: Monogenic cubic cyclic extension of the ring of integers of a quadratic field
zbMATH Open document ID: 0821.11057
DOI: 10.1007/BF01188572
published in: Archiv der Mathematik
publication date: 1995-09-27T00:00:00Z
review text: Let \(A\) be the ring of integers of the quadratic field \(\mathbb{Q} (\sqrt d)\), \(d\neq -3\). In [Arch. Math. 61, No. 4, 348-361 (1993; Zbl 0789.11058)] the author showed that if \(A\) has a cubic, cyclic, monogenic extension of discriminant one, then either of two diophantine equations must have a solution in \(A\). In this paper the author shows that only one of the equations has solutions which yields such extensions: \(A\) has such an extension iff \(4m^ 3= y^ 2 d+27\) has a solution in \(\mathbb{Z}\) with \(d\not\equiv 11, 21, 23, 35\pmod {36}\) or \(d\equiv 1\pmod {12}\) and \(m\not\equiv 3\pmod 9\). The defining polynomial and nontrivial automorphism of the extension are determined.
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 756421
zbMATH Keywords: cubic, cyclic, monogenic extension
zbMATH Keywords: diophantine equations
zbMATH Keywords: defining polynomial
author: Jean-Daniel Therond
reviewed by: Lindsay N. Childs
MaRDI profile type: Publication
cites work: Weitere Ergebnisse über \(x^ 3+y^ 3=z^ 3\) in quadratischen Körpern
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full work available at URL: https://doi.org/10.1007/bf01188572
OpenAlex ID: W395712538
description: scientific article; zbMATH DE number 756421