Lattices of compatibly embedded finite fields
type: Item
instance of: scholarly article
title: Lattices of compatibly embedded finite fields
zbMATH Open document ID: 0895.11055
author: Wieb Bosma
author: John J. Cannon
author: Allan K. Steel
published in: Journal of Symbolic Computation
publication date: 1997-11-24T00:00:00Z
review text: The paper under review discusses the main problems involved in developing a finite field module for a computer algebra system, and describes the solutions to these problems that have been adopted in the MAGMA [\textit{J. Cannon} and \textit{C. Playoust}, Magma: A new computer algebra system, Euromath Bull. 2, 113-144 (1996)] system. Perhaps the most difficult problem is one of compatibility. Suppose \(E\) and \(F\) are finite fields with \(E\) ``contained in'' \(F\) in the sense that \(F\) contains a subfield isomorphic to \(E\). If \(S\) is another finite field explicitly embedded in both \(E\) and \(F\), then it is highly desirably to choose an explicit embedding of \(E\) in \(F\) such that, when an element of \(S\) is expressed as an element of \(F\), it does not matter whether this is done directly or via the intermediate field \(E\). Moreover, it is crucial that this compatibility be maintained as new fields are added to the lattice. Of course, in many applications one wants the possibility of having several fields of the same cardinality. The MAGMA system solves this compatibility problem in a way that involves only elementary linear algebra and the computation of roots of polynomials. In particular, an explicit representation for an embedding of \(E\) in \(F\) is stored as a matrix with entries from the common prime field \(\mathbb{Z}_p\). In terms of representing a finite field itself, MAGMA uses three basic data structures. Prime fields are of course represented by the integers modulo \(p\) with the actual implementation varying with the size of \(p\). Extension fields with cardinality at most \(2^{20}\) are represented as powers of a primitive element together with a table of logarithms used for addition, the so-called Zech representation. Larger extension fields are viewed as an extension of a Zech field, if possible, with elements represented as specialized polynomials over the Zech field. There is one misstatement in the paper noticed by the reviewer. T…
Mathematics Subject Classification ID: 11Y16
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 11T99
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 12E20
zbMATH DE Number: 1077118
zbMATH Keywords: finite fields
zbMATH Keywords: compatibility
zbMATH Keywords: representations
zbMATH Keywords: computer algebra system
reviewed by: Gary L. Ebert
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W1965120124
DOI: 10.1006/JSCO.1997.0138
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description: scientific article; zbMATH DE number 1077118