Elements of Algebra: Geometry, Numbers, Equations

192 Pages·1994·5.29 MB·English

by  John Stillwell

John Stillwell
Elements of Algebra
Geometry, Numbers, Equations
With 34 illustrations
Springer
John Stillwell
Department of Mathematics
Monash University
Clayton, Victoria 3168
Australia
Editorial Board:
S. Axler F.W. Gehring
Department of Mathematics Department of Mathematics
Michigan State University University of Michigan
East Lansing, MI 48824 Ann Arbor, MI 48109
USA USA
P.R. Halmos
Department of Mathematics
Santa Clara University
Santa Clara, CA 95053
USA
Mathematics Subject Classification (1991): 00-01. 08-01, 115XX, 14-XX
-----------------------
Library of Congress Cataloging-in-Publication Data
Stillwell, John.
Elements of algebra: geometry, numbers, equations / John
Stillwell.
p. cm. --(Undergraduate texts in mathematics)
Includes bibliographical references and index.
ISBN 0-387-94290-4. --ISBN 3-540-94290-4
I. Algebra. I. Title. II. Series.
QA155.S75 1994
512' .02-dc20 94-10085
Printed on acid-free paper.
© 1994 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the writ
ten permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY
100 I 0, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in con
nection with any form of information storage and retrieval, electronic adaptation, computer soft
ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Production managed by Karen Phillips, manufacturing supervised by Vincent Scelta.
Photocomposed pages prepared from the author's LATEX file.
Printed and bound by Edwards Brothers, Inc., Ann Arbor, MI.
Printed in the United States of America.
9 8 7 6 5 4 3 2 (Corrected second printing, 1996)
ISBN 0-387-94290-4 Springer-Verlag New York Berlin Heidelberg
ISBN 3-540-94290-4 Springer-Verlag Berlin Heidelberg New York SPIN 10546993
To Elaine

Preface
Algebra is abstract mathematics let us make no bones about it - yet it is
also applied mathematics in its best and purest form. It is not abstraction for its
own sake, but abstraction for the sake of efficiency, power and insight. Algebra
emerged from the struggle to solve concrete, physical problems in geometry, and
succeeded after 2000 years of failure by other forms of mathematics. It did this
by exposing the mathematical structure of geometry, and by providing the tools
to analyse it. This is typical of the way algebra is applied; it is the best and
purest form of application because it reveals the simplest and most universal
mathematical structures.
The present book aims to foster a proper appreciation of algebra by showing
abstraction at work on concrete problems, the classical problems of construction
by straightedge and compass. These problems originated in the time of Euclid,
when geometry and number theory were paramount, and were not solved until
the 19th century, with the advent of abstract algebra. As we now know, alge
bra brings about a unification of geometry, number theory and indeed most
branches of mathematics. This is not really surprising when one has a historical
understanding of the subject, which I also hope to impart.
The bridge between Euclid and abstract algebra is the algebraic geometry
invented by Fermat and Descartes around 1630. By assigning numerical coordi
nates to points in the plane, they were able to restate many geometric problems
as problems about polynomial equations. Thanks to 16th century advances in
the treatment of equations, they found it easy to solve many problems, some
of which had defeated the ancients. However, certain problems remained in
tractable, particularly those involving equations of degree 2: 3. At first it was
thought that improved technique would solve these too, but as time went by
this hope faded, and a fundamental shift in thinking took place. By the end
of the 18th century, mathematicians were considering the possibility of equa
tions without solutions, or at least without solutions of a certain geometric type
("constructible" solutions).
Such a possibility calls for a more abstract level of algebraic thought. Instead
of treating equations, it is necessary to treat properties of equations, since the
goal is to recognise the property of solvability. Since the equations are polynomial
equations, this amounts to studying the properties of polynomials. This was done
very successfully by Lagrange, Gauss, Abel and Galois between 1770 and 1830.
Their work was successful not only in solving the ancient construction problems,
but also in creating the concepts that are the backbone of algebra today - rings,
fields, and groups.
However, these concepts were not very clear at the time. They were identified
and disentangled from the old theory of equations only a century later, through
the work of Jordan, Kronecker, Dedekind, Noether and Artin, and first presented
to the general mathematical public in the Moderne Algebra of van der Waerden
[1931]. The result was an algebra which, ironically, could live without geometry
- and to some mathematicians that meant an algebra which should live without
viii Preface
geometry. Other mathematicians (and students!) were alienated and bewildered
by this development, particularly those who were geometrically inclined. It is true
that algebra was separated from geometry with the best of intentions. Some of
the leading "separatists," in fact, were geometers who wanted to build rigorous
foundations for algebraic geometry and topology, and this called for an algebra
without any geometric assumptions. All the same, separation of algebra from
geometry was a pedagogical mistake, and fortunately one from which we are
beginning to recover.
Since the great virtue of algebra is its power to unify topics from number
theory to geometry, why not develop the subject with unification in mind? I
hope that the present book demonstrates that such a unification is possible. It
grew out of a course in Galois theory for 3rd year students at Monash Univer
sity, but should in principle be accessible to anyone with a strong secondary
school background, assuming this background includes the language of sets and
functions. On the other hand, it should also be of interest to mathematicians
who know the technicalities of abstract algebra but wish to know more about
its historical context. Although it is no substitute for a comprehensive history of
algebra (which has yet to be written), this book does try to locate the sources of
the main ideas. They can be picked up on the fly from the references to the orig
inal literature, in the name [year] format, or mulled over in the end-of-chapter
discussions. Since the aim of the book is to lead the reader to better things, I
hope these discussions will open the door to the great works by Gauss, Abel,
Galois and others. Looking to the future as well as the past, there is also some
discussion of recent developments and open problems.
The book is divided into sections small enough to be digested in one sitting.
There is at most one theorem per section, so theorems can be identified by their
section numbers. For example, Theorem 8.4 refers to the theorem in Section 8.4.
The most frequently used theorems have been given names rather than numbers,
so that it will be easier to recall what they are about. Exercises are placed in
small groups at the end of sections, in the hope that they will be more tempting,
and less difficult, when the reader's mind is already on the right track. They
include many interesting theorems that could not be squeezed into the main
text. The starred sections contain material that can be omitted from a basic
course. If time is short (as it is at Monash, where we have only 24 lectures), then
Chapters 3 and 9 can also be omitted.
I would like to thank Emma Carberry, Angelo di Pasquale, Helena Gregory,
Mark Kisin, Sean Lucy, Greg Pantelides, Karen Parshall, Abe Shenitzer, Tanya
Staley and Drew Vandeth for many corrections and improvements, Anne-Marie
Vandenberg for her usual splendid typing, and my wife Elaine for her sharp-eyed
proofreading.
Clayton, Victoria, Australia John Stillwell
Contents
Preface vii
Chapter 1. Algebra and Geometry 1
1.1 Algebraic Problems 1
1.2 Straightedge and Compass Constructions 1
1.3 The Constructible Numbers 5
lA Some Famous Constructible Figures 7
1.5 The Classical Construction Problems 10
1.6 Quadratic and Cubic Equations 11
,1.7 Quartic Equations 13
1.8 Solution by Radicals 14
1.9 Discussion 15
Chapter 2. The Rational Numbers 18
2.1 Natural Numbers 18
2.2 Integers and Rational Numbers 20
2.3 Divisibility 22
2.4 The Euclidean Algorithm 23
2.5 Unique Prime Factorisation 25
2.6 Congruences 26
2.7 Rings and Fields of Congruence Classes 28
2.8* The Theorems of Fermat and Euler 29
2.9* Fractions and the Euler Phi Function 32
2.10 Discussion 34
Chapter 3. Numbers in General 38
3.1 Irrational Numbers 38
3.2 Existence and Meaning of Irrational Numbers 40
3.3 The Real Numbers 41
3.4 Arithmetic and Rational Functions on R. 42
3.5 Continuity and Completeness 44
3.6 Complex Numbers 46
3.7 Regular Polygons 48
3.8 The Fundamental Theorem of Algebra 51
3.9 Discussion 53
Chapter 4. Polynomials 57
4.1 Polynomials over a Field 57
4.2 Divisibility 58
4.3 Unique Factorisation 61
4.4 Congruences 62
x Contents
4.5 The Fields F(a) 63
4.6 Gauss's Lemma 65
4.7 Eisenstein's Irreducibility Criterion 67
4.8* Cyclotomic Polynomials 69
4.9* Irreducibility of Cyclotomic Polynomials 71
4.10 Discussion 73
Chapter 5. Fields 76
5.1 The Story So Far 76
5.2 Algebraic Numbers and Fields 76
5.3 Algebraic Elements over an Arbitrary Field 77
5.4 Degree of a Field over a Subfield 78
5.5 Degree of an Iterated Extension 81
5.6 Degree of Constructible Numbers 83
5.7* Regular n-gons 85
5.8 Discussion 86
Chapter 6. Isomorphisms 89
6.1 Ring and Field Isomorphisms 89
6.2 Isomorphisms of Q(a) and F(a) 91
6.3 Extending Fields and Isomorphisms 94
6.4 Automorphisms and Groups 97
6.5* Function Fields and Symmetric Functions 98
6.6* Cyclotomic Fields 100
6.7* The Chinese Remainder Theorem 101
6.8 Homomorphisms and Quotient Rings 103
6.9 Discussion 105
Chapter 7. Groups 108
7.1 Why Groups? 108
7.2 Cayley's Theorem 109
7.3 Abelian Groups 111
7.4 Dihedral Groups 112
7.5* Permutation Groups 115
7.6* Permutation Groups in Geometry 116
7.7 Subgroups and Cosets 119
7.8 Normal Subgroups 121
7.9 Homomorphisms 122
7.10 Discussion 125
Chapter 8. Galois Theory of Unsolvability 128
8.1 Galois Groups 128
8.2 Solution by Radicals 130
8.3 Structure of Radical Extensions 132
Contents xi
8.4 Nonexistence of Solutions by Radicals when n ~ 5 134
8.5* Quintics with Integer Coefficients 136
8.6* Unsolvable Quintic Equations with Integer Coefficients 138
8.7* Primitive Roots 139
8.8 Finite Abelian Groups 141
8.9 Discussion 143
Chapter 9. Galois Theory of Solvability 146
9.1 The Theorem of the Primitive Element 146
9.2 Conjugate Fields and Splitting Fields 148
9.3 Fixed Fields 150
9.4 Conjugate Intermediate Fields 152
9.5 Normal Extensions with Solvable Galois Group 154
9.6 Cyclic Extensions 155
9.7 Construction of the Radical Extension 156
9.8 Construction of Regular p-gons 157
9.9* Division of Arbitrary Angles 159
9.10 Discussion 160
References 162
Index 170