On the resolution of index form equations in sextic fields with an imaginary quadratic subfield
type: Item
instance of: scholarly article
title: On the resolution of index form equations in sextic fields with an imaginary quadratic subfield
zbMATH Open document ID: 0873.11025
author: István Gaál
author: Michael E. Pohst
published in: Journal of Symbolic Computation
publication date: 1997-10-30T00:00:00Z
review text: Let \(K\) be an algebraic number field with ring of integers \(\mathbb{Z}_K\). The index of the primitive integer \(\alpha \in\mathbb{Z}_K\) is defined as the module index \(I(\alpha) = [\mathbb{Z}_K: \mathbb{Z}[a]]\). Solving the equation \(I(\alpha) =\pm 1\) is then equivalent to finding all primitive \(\alpha\in \mathbb{Z}_K\) such that \(1,\alpha, \alpha^2, \dots\) is an integral basis (i.e. a \(\mathbb{Z}\)-basis for \(\mathbb{Z}_K)\). The authors consider the case in which \(K\) is a sextic field containing an imaginary quadratic subfield \(M\). In very many cases an integer \(\theta \in \mathbb{Z}_K\) can be found such that \(\theta\) has relative index 1 over \(M\). Under this extra assumption on the sextic field \(K\), the authors develop an algorithm for solving index form equations \(I(\alpha) =\pm 1\). They show first that such index form equations are reduced to certain cubic Thue equations with integral unknowns in \(M\). These cubic Thue equations are solved, in turn, by a method that makes use of linear forms in logarithms of algebraic numbers, very similar to that of solving ordinary Thue equations, in three stages: (i) Application of a Baker-type theorem (in this case the theorem of Baker and Wüstholz) in order to obtain an upper bound (a very large one) for certain unknown exponents, (ii) Reduction of the upper bound using the LLL basis reduction algorithm and (iii) Enumeration of all solutions below the reduced upper bound; conceptually, a trivial stage, but the most time consuming. Lots of numerical examples are given.
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11Y50
Mathematics Subject Classification ID: 11Y40
zbMATH DE Number: 987589
zbMATH Keywords: LLL basis reduction algorithm
zbMATH Keywords: Baker's method
zbMATH Keywords: power integral basis
zbMATH Keywords: sextic field containing an imaginary quadratic subfield
zbMATH Keywords: algorithm
zbMATH Keywords: index form equations
zbMATH Keywords: cubic Thue equations
zbMATH Keywords: linear forms in logarithms
reviewed by: Nikos Tzanakis
MaRDI profile type: Publication
full work available at URL: jsco.1996.0060
OpenAlex ID: W1988005548
DOI: 10.1006/JSCO.1996.0060
Recommended article: On the resolution of index form equations in quartic number fields
Recommended article: On the Resolution of Index Form Equations in Dihedral Quartic Number Fields
Recommended article: Index form equations in sextic fields: a hard computation
Recommended article: On the resolution of index form equations in biquadratic number fields. II
Recommended article: On the resolution of index form equations in biquadratic number fields. I
Recommended article: On the resolution of index form equations in biquadratic number fields. III: The bicyclic biquadratic case
Recommended article: The Computation of Sextic Fields with a Quadratic Subfield
Recommended article: On the solution of units and index form equations in algebraic number fields
Recommended article: Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields
Recommended article: Q5478060
description: scientific article; zbMATH DE number 987589