Simplification of expressions involving radicals
type: Item
instance of: scholarly article
title: Simplification of expressions involving radicals
zbMATH Open document ID: 0574.12002
DOI: 10.1016/S0747-7171(85)80014-6
author: Richard Zippel
published in: Journal of Symbolic Computation
publication date: 1985-01-01T00:00:00Z
review text: These two closely related (and interactively written) papers deal with the problem of ''denesting'' (decreasing the depth of) a nested radical, e.g., \(\sqrt{(5+2\sqrt{6})}=\sqrt{2}+\sqrt{3}\), or \(\sqrt{(4+3\sqrt{2})}=^ 4\sqrt{2}+^ 4\sqrt{8}\), or \({}^ 4\sqrt{- 4}=1+\sqrt{-1}\), etc. There are many more provocative curiosities, e.g., see \textit{D. Shanks} [Fibonacci Q. 12, 271, 280 (1974; Zbl 0294.12001)], including combinations of radicals which denest collectively but not individually. This is a matter for symbolic computation since we must identify equivalent answers by machine. In principle, this requires discovering a (possibly larger) field in which to use a rational basis to treat radicals as polynomials and also a simplification process (because the most standard form might not be the most easily recognized). The complexity seems generally exponential. The first paper deals with the quadratic case. With restriction to reals only the fourth root can enter additionally. A program is given for denesting; an essential feature is the recognition of perfect squares (e.g., of the norms of the radicands). The unpleasant surprise is that the nest of depth n is resolved into two nests of depth n-1, thus increasing the depth when \(n>2.\) The second paper deals more theoretically with Kummer extensions, using illustrations of degree 2 and 3. As before, we must first discover the larger working field. The earliest reference is to \textit{A. Capelli} [Rend. Accad. Sci. Fis. Mat., Napoli, III. Ser. 3, 243-252 (1897))]. The almost simultaneous Hilbert Zahlbericht escapes notice, but not the Artin-Tate cohomology, which is used in an appendix to find the Galois group of a Kummer field. Work of \textit{M. Kneser} [Acta. Arith. 26, 307- 308 (1975; Zbl 0314.12001)] and \textit{F. Halter-Koch} [J. Algebra 63, 318- 330 (1980; Zbl 0428.12020)] is cited to show that non-Kummerian extensions (with insufficient roots of unity) may also be used.
Mathematics Subject Classification ID: 12-04
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 3916397
zbMATH Keywords: exponential complexity
zbMATH Keywords: nested radical
zbMATH Keywords: symbolic computation
zbMATH Keywords: Kummer extensions
Wikidata QID: Q56171822
reviewed by: Harvey Cohn
MaRDI profile type: Publication
cites work: Q5566638
cites work: On the Linear Independence of Fractional Powers of Integers
cites work: On Canonical Forms and Simplification
cites work: Sur quelques points d'algèbre homologique
cites work: Eine Galoiskorrespondenz für Radikalerweiterungen
cites work: Lineare Abhängigkeit von Wurzeln
cites work: Q5622122
cites work: Q5542886
cites work: Q5513003
cites work: Q3267461
cites work: An application of Galois theory to elementary arithmetic
cites work: Factoring Polynomials Over Algebraic Number Fields
cites work: Profinite Groups, Arithmetic, and Geometry. (AM-67)
cites work: Q3947744
cites work: Q5789039
cites work: Factoring Multivariate Polynomials over Algebraic Number Fields
DBLP publication ID: journals/jsc/Zippel85
description: scientific article; zbMATH DE number 3916397