Beyond the quartic equation
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 subject. On page~21, lines~1--3, he writes: ``A characteristic feature of the symmetric permutation group \(S_n\) for all \(n\) is that all permutations having the same cycle structure come from the same conjugacy class.'' For the proof of this assertion he refers to a book on group theoretical techniques in Quantum Chemistry. On the one hand the assertion about \(S_n\) is immediately clear from the form of the conjugacy operation in \(S_n\); on the other hand it is true for normal subgroups of \(S_n\) as well. There are some other places in the text that might astonish an experienced reader, but in general the book can be recommended to anyone interested in the solution of quintic equations. [For a review of the original edition, published by Birkhäuser in 1996, see Zbl 0905.12002.]
scientific article; zbMATH DE number 5359364