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Beyond the Quartic Equation

R. Bruce King
Publication Date: 
Number of Pages: 
Modern Birkhäuser Classics
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Art Gittleman
, on

Various books develop the theory of equations, usually including some Galois theory and the conditions for solvability by radicals. But what of equations that are not solvable by radicals, which are of degree beyond the quartic? The author found the late 19th century work, Kiepert's algorithm, that provides the basis for an explicit solution of the quintic. He enlisted the help of computer scientist E. R. Canfield to write a computer program that would calculate the roots of a quintic from its coefficients using this algorithm. King and Canfield published the algorithm in 1991 but the whole story was too long to tell in a journal article.

This short book of 149 pages does read more like a journal article than a textbook. There are few examples and no exercises or computations. The author's goal is to make the material available to non-specialists and mathematically oriented readers, but is unsatisfying in that it has too much introductory material for someone who has seen this material before and not enough for someone who has not.

For example, in Chapter 2, on "Group Theory and Symmetry," a group is defined and then the properties of conjugate elements are listed without explanation or proof. Later in that chapter the representation of complex numbers in the plane is described and the Riemann sphere is mentioned without explaining the stereographic projection. As in a journal article, there is no index, so it is hard to determine if the polyhedral polynomials discussed here are referred to later.

After defining rings, integral domains, and fields, Chapter 3 introduces Galois theory and Tschirnhausen transformations. Chapter 4 present elliptic functions as a generalization of radicals and covers theta functions, which can be used to compute values of elliptic functions. Chapter 5, Algebraic Equations Soluble by Radicals, might have been switched with Chapter 4. Chapter 6 is the heart of the book, covering the Kiepert algorithm for the roots of the general quintic. This involves several transformations of the equation, solution by Weierstrass elliptic functions, and evaluation by theta functions. For completeness the final two chapters present the methods of Hermite and Gordan and the sextic and septic equations briefly.

Mathematica has a nice implementation of the quintic solution and even has a poster that displays it. But Beyond the Quartic is a nice source for an explanation and clarification of Kiepert's 1878 work.

Art Gittleman ( is Professor of Computer Science at California State University Long Beach. 




Group Theory and Symmetry

The Symmetry of Equations: Galois Theory and Tschirnhausen Transformations

Elliptic Functions

Algebraic Equations Soluble by Radicals

The Kiepert Algorithm for Roots of the General Quintic Equation

The Methods of Hermite and Gordan for Solving the General Quintic Equation

Beyond the Quintic Equation