On computing subfields
type: Item
instance of: scholarly article
title: On computing subfields
zbMATH Open document ID: 0886.11072
published in: Journal of Symbolic Computation
publication date: 1997-11-25T00:00:00Z
review text: [This is a joint review for three papers (see also Zbl 0886.11070 and Zbl 0886.11071)] The first paper (Daberkow et al., Zbl 0886.11070) reads like seven computer salesmen presenting KANT V4. The new idea is to call on algorithms to solve problems of relative subfields without first having to select a subset of the costly computations over \(\mathbb Q\). In 1970, fields of degree \(>4\) were out of reach, while in 1990, fields of degree \(>10\) were still out of reach. Now degree 20 and higher (maybe 1000) seems accessible. The Zassenhaus objectives are again embraced (viz., to study orders, units, class numbers, and Galois groups). The algorithms come from MAGMA through the now traditional sources [\textit{H. Cohen}, A course in computational algebraic number theory, Grad. Texts Math. 138, Springer (1993; Zbl 0786.11071), and \textit{M. Pohst} and \textit{H. Zassenhaus}, Algorithmic algebraic number theory, Encycl. Math. Appl. 30, Cambridge University Press (1989; Zbl 0685.12001)]. The shell KASH is advertised through enticing sample inputs and outputs. The use of subfield techniques helps in finding Hilbert class fields (by Kummer extensions), solving relative norm problems, and in Galois group techniques for listing subfields. There are some hints of pure mathematical interest, for example the use of approximations to \(L\)-series to find a lower bound on the class number to use with an upper bound obtained by the orders of individual primes (with the help of GRH). The most intriguing item might be the solution of Thue's equation, which is unexplained together with an arcane reference [\textit{Y. Bilu} and \textit{G. Hanrot}, Solving Thue's equations of high degree, Mathematica Gottingensis, Schriftenreihe des Sonderforschungsbereichs Geometrie und Analysis, Heft 11, Göttingen (1995), (added in 2013) final version in J. Number Theory 60, No. 2, 373--392 (1996; Zbl 0867.11017)]. The second paper (Daberkow, Zbl 0886.11071) presents some of the details of the use of…
reviewed by: Harvey Cohn
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 1077120
zbMATH Keywords: Hensel lifting
zbMATH Keywords: Hilbert class fields
zbMATH Keywords: Galois groups
author: Jürgen Klüners
author: Michael E. Pohst
describes a project that uses: KANT/KASH
MaRDI profile type: Publication
full work available at URL: jsco.1996.0140
OpenAlex ID: W1996725012
DOI: 10.1006/JSCO.1996.0140
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description: scientific article; zbMATH DE number 1077120