Beyond the quartic equation
type: Item
instance of: scholarly article
title: Beyond the quartic equation
zbMATH Open document ID: 1177.12001
author: B. E. Eshmatov
published in: Modern Birkhäuser Classics
publication date: 2008-10-30T00:00:00Z
review text: The idea of the book at hand is the development of a practicable algorithm to solve quintic equations by means of elliptic and theta functions. It was \textit{Ch.~Hermite} [C. R. Acad. Sci. Paris 46 (1858)] who first showed that elliptic functions provide solutions to the general quintic equation. This idea was developed further by several authors including \textit{L.~Kiepert} [J. Reine Angew. Math. 87, 114--133 (1879; JFM 11.0074.02; JFM 10.0073.04)] and \textit{F.~Klein} [``Vorlesungen über das Ikosaeder und die Auf{\null}lösung der Gleichungen vom fünften Grade.'' Teubner, Leipzig (1884, JFM 16.0061.01)]. The author uses the methods of Kiepert to devise a computer program which delivers the solution of quintic equations. The first step is the reduction of the general quintic equation to a one-parameter equation by means of Tschirnhausen transformations. The perhaps easiest one-parameter equation is \(x^5 + ax + a = 0\). But the Tschirnhausen transformation to this normal form requires the use of two square roots and one root of a cubic equation. Kiepert uses the normal form \(x^5 - 10bx^3 + 45b^2x - b^2 = 0\). The corresponding Tschirnhausen transformation requires only two square roots. In the second step one solves the normal equation by means of Weierstraß elliptic functions and theta functions. The book consists of eight chapters: 1.~Introduction, 2.~Group Theory, 3.~Galois Theory, 4.~Elliptic Functions, 5.~Algebraic Equations Soluble by Radicals, 6.~The Kiepert Algorithm for Roots of the General Quintic Equation, 7.~The Methods of Hermite and Jordan for Solving the General Quintic Equation, 8.~Beyond the Quintic Equation. The author wants to make the key ideas of the Kiepert algorithm accessible to nonspecialists. Therefore he explains the necessary background in the chapters 1 to~4. The reviewer wishes to conclude with some critical remarks. The author always writes ``Cardan'' instead of ``Cardano''. He is far from giving a historical introduction to the su…
reviewed by: Helmut Koch
Mathematics Subject Classification ID: 12-03
Mathematics Subject Classification ID: 12E12
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 12-02
zbMATH DE Number: 5359364
zbMATH Keywords: quintic equations
zbMATH Keywords: elliptic and theta functions
zbMATH Keywords: Tschirnhausen transformations
zbMATH Keywords: methods of Kiepert
zbMATH Keywords: Kiepert algorithm
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5359364