Gröbner bases applied to finitely generated field extensions
type: Item
instance of: scholarly article
title: Gröbner bases applied to finitely generated field extensions
zbMATH Open document ID: 0997.12005
author: Joern Mueller-Quade
author: Rainer Steinwandt
published in: Journal of Symbolic Computation
publication date: 2002-11-11T00:00:00Z
review text: In this article, the authors address practical problems in the computation with finitely generated field extensions. In particular, they give efficient algorithms to compute the (transcendence) degree of such an extensions, decide separability in positive characteristic, decide normality, give a field-membership test and present methods allowing the computation of minimal polynomials over subfields. The main technique used is that of Gröbner basis computations. This paper continues previous work of the authors [J. Symb. Comput. 27, 143-170 (1999; Zbl 0935.11046)]. The main difference between this approach and older ideas by \textit{M. E. Sweedler} [Lect. Notes Comput. Sci. 673, 66-75 (1993; Zbl 0813.13026)] and \textit{G. Kemper} [An algorithm to determine properties of field extensions lying over a ground field, IWR preprint 93-58, Heidelberg, Germany (1993)] is to avoid the use of tag-variables as they tend to slow down the necessary Gröber basis computations considerably. The article gives several worked out examples using Magma [\textit{W. Bosma, J. Cannon} and \textit{C. Playoust}, J. Symb. Comput. 24, 235-265 (1997; Zbl 0898.68039)] demonstrating the efficiency of their new methods, e.g. they verify the invariant field of a group of order \(248\) that was given in \textit{G. Kemper}'s PhD thesis [Noether's problem and generic polynomials (in German). Heidelberg (1994; Zbl 0830.12003)].
reviewed by: Claus Fieker
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 1538754
zbMATH Keywords: algorithms for field extensions
zbMATH Keywords: Gröbner bases
zbMATH Keywords: transcendence degree
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W2029267855
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DOI: 10.1006/JSCO.1999.0417
description: scientific article; zbMATH DE number 1538754