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Modern Birkhäuser Ciassics Modern Birkhäuser CIassics Many of the original research and survey monographs in pure and applied mathematics published by Birkhäuser in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain accessible to new generations of students, scholars, and researchers. Beyond the Quartic Equation R. Bruce King Reprint of the 1996 Edition Birkhauser Boston • Basel • Berlin R. Bruce King Department of Chemistry University of Georgia Athens, GA 30602 USA [email protected] ISBN: 978-0-8176-4836-7 e-ISBN: 978-0-8176-4849-7 DOI: 10.1007/978-0-8176-4849-7 Library of Congress Control Number: 2008938133 Mathematics Subject Classification (2000): 14H42, 14K25, 11Rxx, 11Sxx ©Birkhauser Boston, a part of Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY, 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper birkhauser.com R. Bruce King Beyond the Quartic Equation Birkhauser Boston*Basel#Berlin R. Bruce King Department of Chemistry University of Georgia Athens, GA 30602 Library of Congress Cataloging-In-Publication Data King, R. Bruce Beyond the quartic equation / R. Bruce King. p. cm. Includes bibliographical references. ISBN 0-8176-3776-1 (hardcover : acid-free). - ISBN 3-7643-3776-1 1. Quintic equations. I. Title. QA215.K48 1996 512.9'42--dc20 96-22841 CIP Printed on acid-free paper fi&® © 1996 Birkhauser Boston Birkhauser W) Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 0-8176-3776-1 ISBN 3-7643-3776-1 Camera-ready text prepared by the Author in Microsoft Word. Printed and bound by Quinn-Woodbine, Woodbine, NJ. Printed in the U.S.A. 98765432 Contents Preface ix 1. Introduction 1 2. Group Theory and Symmetry 6 2.1 The Concept of Groups 6 2.2 Symmetry Groups 9 2.3 Regular Polyhedra 13 2.4 Permutation Groups 19 2.5 Polyhedral Polynomials 26 2.6 Transvectants of Polyhedral Polynomials 30 3. The Symmetry of Equations: Galois Theory and Tschirnhausen Transformations 34 3.1 Rings, Fields, and Polynomials 34 3.2 Galois Theory: Solubility of Algebraic Equations by Radicals . 45 3.3 Tschirnhausen Transformations 51 4. Elliptic Functions 56 4.1 Elliptic Functions by the Generalization of Radicals 56 4.2 Elliptic Functions as Doubly Periodic Functions 62 4.3 Theta Functions 72 4.4 Higher Order Theta Functions 79 5. Algebraic Equations Soluble by Radicals 82 5.1 The Quadratic and Cubic Equations 82 5.2 The Quartic Equation 87 5.3 Special Quintic Equations Solvable by Radicals 89 6. The Kiepert Algorithm for Roots of the General Quintic Equation . 95 6.1 Introduction 95 6.2 The Tschirnhausen Transformation of the General Quintic to the Principal Quintic 99 6.3 The Tschirnhausen Transformation of the Principal Quintic to the Brioschi Quintic 103 6.4 Transformation of the Brioschi Quintic to the Jacobi Sextic . 108 6.5 Solution of the Jacobi Sextic with Weierstrass Elliptic Functions 113 6.6 Evaluation of the Weierstrass Elliptic Functions Using Genus 1 Theta Functions 118 6.7 Evaluation of the Periods of the Elliptic Functions Corresponding to the Jacobi Sextic 123 viii Contents 6.8 Undoing the Tschirnhausen Transformations 126 7. The Methods of Hermite and Gordan for Solving the General Quintic Equation 128 7.1 Hermite's Early Work on the Quintic Equation 128 7.2 Gordan's Work on the Quintic Equation 132 8. Beyond the Quintic Equation 139 8.1 The Sextic Equation 139 8.2 The Septic Equation 143 8.3 The General Algebraic Equation of any Degree 147 Preface One of the landmarks in the history of mathematics is the proof of the nonexis- tence of algorithms based solely on radicals and elementary arithmetic operations (addition, subtraction, multiplication, and division) for solutions of general alge- braic equations of degrees higher than four. This proof by the French mathemati- cian Evariste Galois in the early nineteenth century used the then novel concept of the permutation symmetry of the roots of algebraic equations and led to the invention of group theory, an area of mathematics now nearly two centuries old that has had extensive applications in the physical sciences in recent decades. The radical-based algorithms for solutions of general algebraic equations of degrees 2 (quadratic equations), 3 (cubic equations), and 4 (quartic equations) have been well-known for a number of centuries. The quadratic equation algorithm uses a single square root, the cubic equation algorithm uses a square root inside a cube root, and the quartic equation algorithm combines the cubic and quadratic equation algorithms with no new features. The details of the formulas for these equations of degree d(d = 2,3,4) relate to the properties of the corresponding symmetric groups Sd which are isomorphic to the symmetries of the equilateral triangle for d = 3 and the regular tetrahedron for d — 4. Related ideas can be used to generate an algorithm for solution of the general algebraic equation of degree 5 (the quintic equation). Such a quintic equation algorithm does not violate the classical theorem proved by Galois since it contains more complicated mathematical functions than the radicals that suffice for the algorithms for roots of the general quadratic, cubic, and quartic equations. The underlying mathematics for an algorithm to solve the quintic equation was developed by nineteenth century European mathematicians shortly after Galois' discovery of the insolubility of the general quintic equation using only radicals. The initial work in this area was done by Hermite and then developed further by Gordan. This culminated in two key publications, an 1878 article by Kiepert1 describing a quintic equation algorithm and the classic 1884 book by Klein2 de- scribing the relationship between the icosahedron and the solution of the quintic equation. At that stage this work lay fallow for more than a century since the algorithm for roots of the general quintic equation appeared intractable before the era of computers. Many of the key ideas appear to have been forgotten by the subsequent generations of mathematicians during the past century so that some of the underlying mathematics has the status of a lost art. The work discussed in this book arose when I discovered the 1878 Kiepert paper and wished to see if the quintic equation algorithm described therein would work on a modern computer. I enlisted the help of a computer scientist, Prof. E. R. Canfield of the University of Georgia, to write a program that would use the Kiepert algorithm to solve the quintic equation (i.e., to calculate the roots of the quintic equation from its coefficients). This proved to be much more difficult than anticipated because of errors and lack of detail in the Kiepert paper. However, X Preface the quintic equation algorithm was finally made to work on an IBM-compatible personal computer but only after studying a considerable amount of related math- ematics in several languages mainly in nineteenth century journals. This work led to the discovery of some unanticipated features of the quintic equation algorithm. This book presents for the first time a complete algorithm for the roots of the general quintic equation with enough background information to make the key ideas accessible to nonspecialists and even to mathematically oriented readers who are not professional mathematicians. Two relatively short papers3,4, have been published on this work on the quintic equation algorithm but the whole story is far too long to fit into a journal article of reasonable length. The book includes initial introductory chapters on group theory and symmetry, the Galois theory of equations, and some elementary properties of elliptic functions and associated theta functions in an attempt to make some of the key ideas accessible to less sophisticated readers. The book also includes discussion of the much simpler algorithms for roots of the general quadratic, cubic, and quartic equations before discussing the algorithm for the roots of the general quintic equation. The book concludes with a brief discussion of attempts to extend these ideas to algorithms for the roots of general equations of degrees higher than five. I am indebted to Prof. E. R. Canfield of the Computer Science Department at the University of Georgia for helpful discussions during the varous stages of this project.
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