Computing elements of given index in totally complex cyclic sextic fields
type: Item
instance of: scholarly article
title: Computing elements of given index in totally complex cyclic sextic fields
zbMATH Open document ID: 0857.11068
published in: Journal of Symbolic Computation
publication date: 1996-06-06T00:00:00Z
review text: The author develops a method for solving index form equations in totally complex sextic fields \(K\) which are composites of an imaginary quadratic subfield \(M\) and a totally real cubic subfield \(L\). In a first step the index form equation for \(K\) is reduced to finitely many systems of two norm equations, one from \(K\) over \(M\), the other one from \(L\) over \(\mathbb{Q}\). In a second step the relative norm equations are reduced to cubic Thue inequalities over \(\mathbb{Z}\). The method is then used to demonstrate that in the composite of \(L = \mathbb{Q} (\vartheta_a)\) (for a root \(\vartheta_a\) of \(x^3 - ax^2 - (a+3) x+1=0)\) and \(M = \mathbb{Q} (\omega) \) \((\omega\) generating the maximal order of the imaginary quadratic field \(\mathbb{Q}(\sqrt {-m}))\), the order \(\mathbb{Z} [\vartheta_a, \omega]\) has no power integral basis for \(a \geq 3\) and \(m \geq 19\).
Mathematics Subject Classification ID: 11Y50
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 871136
zbMATH Keywords: index form equations
zbMATH Keywords: totally complex sextic fields
zbMATH Keywords: norm equations
zbMATH Keywords: cubic Thue inequalities
author: István Gaál
reviewed by: Michael E. Pohst
MaRDI profile type: Publication
full work available at URL: jsco.1995.1038
OpenAlex ID: W2080827099
DOI: 10.1006/JSCO.1995.1038
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description: scientific article; zbMATH DE number 871136