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Abel's Proof
An Essay on the
Sources and Meaning
ofMathematical
Unsolvability
Peter Pesic
The MIT Press
Cambridge, Massachusetts
London, England
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2003 Massachusetts Institute ofTechnology
All rights reserved. No part ofthis book may be reproduced in any form by
any electronic or mechanical means (including photocopying, recording, or
information storage and retrieval) without permission in writing from the
publisher.
This book was set in Palatino by Interactive Composition Corporation and
was printed and bound in the United States ofAmerica.
Permission for the use of the figures has been kindly given by the follow
ing: Dover Publications (figures 2.1, 3.1); Conservation Departmentale de
Musees de Vendee (figure 2.2); Department ofMathematics, University of
Oslo, Norway (figure 6.1); Francisco Gonzalez de Posada (figure 10.1, which
appeared in Investigaci6ny Ciencia,July 1990, page 82); Robert W. Gray (fig-ure 8.3); Lucent TechnologIes, Inc.! Bell Labs (figure 10.2); NatIOnal LIbrary
ofNorway, Oslo Division (figure 10.3).
I am grateful to Jean Buck (Wolfram Research), Peter M. Busichio and
Edward J Eckert (Bell I abs/I llCent Technology), Judy Feldmann, Chry-
seis Fox, John Grafton (Dover Publications), Thomas Hull, Nils Klitkou
(National Library ofNorway, Oslo Division), Purificaci6n Mayoral (Investi-
gaci6ny Ciencia),George Nichols, Lisa Reeve, Yngvar Reichelt (Department
ofMathematics, University ofOslo), Mary Reilly, and Ssu Weng for their
help with the figures. Special thanks to Wan-go Weng for his calligraphy of
the Chinese character ssu (meaning "thought") on the dedication page.
Library of Congress Cataloging-in-Publication Data
Pesic, Peter.
Abel's proof: an essay on the sources and meaning of mathematical
unsolvability / Peter Pesic.
p. cm.
Includes bibliographical references and index.
ISBN 0-262-16216-4 (hc : alk. paper)
1. Equations, roots of. 2. Abel, Niels Henrik, 1802-1829. 1. Title.
QA212 .P47 2003
512.9'4--dc21 2002031991
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Contents
Introduction 1
1 The Scandal ofthe Irrational 5
2 Controversy and Coefficients 23
3 Impossibilities and Imaginaries 47
4 Spirals and Seashores 59
5 Premonitions and Permutations 73
6 Abel's Proof 85
Abel and Galois 95
8 Seeing Symmetries 111
9 The OrderofThings 131
10 Solving the Unsolvable 145
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Vlll Contents
Appendix A: Abel's 1824Paper 155
Appendix B: Abel on the GeneralFonn ofan AlgebraicSolution 171
Appendix C: Cauchy's Theorem on Permutations 175
Notes 181
Acknowledgments 203
Index 205
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1The Scandal ofthe
Irrational
The story begins with a secret and a scandal. About 2,500
years ago, in Greece, a philosopher named Pythagoras and
his followers adopted the motto "All is number." The
Pythagorean brotherhood discovered many important math
ematical tr aths and explored the ways they were manifest in
the world. Butthey also wrapped themselves inmystery, con
sidering themselves guardians of the secrets of mathematics
from the profane world. Because of their secrecy, many de
tails of their work are lost, and even the degree to which they
were indebted to prior discoveries made inMesopotamia and
Egypt remains obscure.
Those who followed looked backon the Pythagoreans as
the source of mathematics. Euclid's masterful compilation,
The Elements, written several hundred years later, includes
Pythagorean discoveries along with later work, culminating
in the construction of the five "Platonic solids," the only solidfigures that are regular (having identical equal-sided faces):
the tetrahedron, the cube, the octahedron, the dodecahedron,
and the icosahedron (figure 1.1). The major contribution of
the Pythagoreans, though, was the concept of mathematical
proof, the idea that one could construct irrefutable demon
strations of theoretical propositions that would admit ofno
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2 Controversy andCoefficients
In his attempt to survey the irrational magnitudes, Euclid
was laying out possible solutions ofwhat we now call the
quadratic equation. But that statement distorts his own way
ofunderstanding these matters and thus is seriously mis
leading. The ancient Greek mathematicians did not know the
word IIalgebra," or the symbolic system we mean by it, even
though many of their propositions can be translated into al-
gebraic language. Euclid kept numbers rigorously separatefrom lines and magnitudes. He might have been horrified by
the way algebra mixes rational and irrational, numbers and
roots. So, as we prepare to consider the meaning of equations,
we need to pause and assess the significance of the mathemat
ical revolution reflected in the rise of algebra as we know it.
The word IIalgebra" is Arabic, not Greek, and refers to
the joining together ofwhat is broken. Thus, inDon Quixote,
Cervantes calls someone who sets broken bones an algebrista.The original Arabic algebraists created recipes for solving
different kinds ofproblems, usually imTolving whole num-
bers. They worked by presenting examples with solutions,
but without the methodical symbolism we now associate
with algebra and without the systematic processes of solution
that are equally important. They did recognize the logical
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3 Impossibilities andImaginaries
Van Roomen was so impressed by Viete's astonishing feat
that he traveled to France expressly to meet him. Here was
a man w 0 c aIme to ave oun t e master art t at wou
solve any algebraic problem. Was it really possible that Viete
had the tools to solve every possible equation, ofwhatever
degree? In the case ofvan Roomen's equation of the forty-
fifth degree, Viete was lucky, for he recognized that par-
ticular equation as closely related to formulas he found in
trigonometry. Had van Roomen changed a single coefficient,
Viete's solution would no longer have been valid. Clearly,
van Roomen had not been envisaging the general problem
of solving equations of higher degrees but had come upon
that specific equation in the course of his trigonometric stud-
ies' and was testing to see if others had discovered it also.
Thus, the solution of van Roomen's equation did not
after all illuminate the general question ofwhether all equa-
tions are solvable in radicals, and indeed Viete's solution in-
volved trigonometry, which (as we shall see) goes beyond
radicals. The tenor of Viete's conviction was clearly that all
to solve the quadratic, cubic, and quartic equations. Other
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