Algebraic number theory and Fermat’s last theorem

334 Pages·2002·8.493 MB·English

by  Tall, David Orme, Stewart, Ian

Algebraic Number Theory
and Fermat's Last Theorem
Third Edition

Algebraic Number Theory
and Fermat's Last Theorem
Third Edition
Ian Stewart
Mathematics Institute
University of Warwick
David Tall
Mathematics Education Research Centre
Institute of Education
University of Warwick
C CRC Press
CS
AP.(cid:9) Taylor &Francis Group
Boca Raton London New York
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Library of Congress Cataloging-in-Publication Data
Stewart, Ian, 1945-
Algebraic number theory and fermat's last theorem / Ian Stewart, David Tall.— 3rd ed.
p. cm.
Rev. ed. of: Algebraic number theory. 2nd. 1987.
Includes bibliographical references and index.
ISBN 1-56881-119-5
1. Algebraic number theory. 2. Fermat's last theorem. I. Tall, David Orme. II. Stewart,
Ian, 1945- Algebraic number theory. III. Title.
QA247 .S76 2001
512'.74—dc21
2001036049
To Ronnie Brown whose brainchild it was

Contents
Preface xi
Index of Notation xvii
The Origins of Algebraic Number Theory 1
I Algebraic Methods 7
I Algebraic Background 9
1.1 Rings and Fields (cid:9) 10
1.2 Factorization of Polynomials (cid:9) 13
1.3 Field Extensions (cid:9) 20
1.4 Symmetric Polynomials (cid:9) 22
1.5 Modules (cid:9) 25
1.6 Free Abelian Groups (cid:9) 26
1.7 Exercises (cid:9) 32
2 Algebraic Numbers 35
2.1 Algebraic Numbers (cid:9) 36
2.2 Conjugates and Discriminants (cid:9) 38
2.3 Algebraic Integers (cid:9) 42
2.4 Integral Bases (cid:9) 45
2.5 Norms and Traces (cid:9) 49
2.6 Rings of Integers (cid:9) 51
2.7 Exercises (cid:9) 57
vii
viii(cid:9) Contents
3 Quadratic and Cyclotomic Fields(cid:9) 61
3.1 Quadratic Fields (cid:9) 61
3.2 Cyclotomic Fields (cid:9) 64
3.3 Exercises (cid:9) 69
4 Factorization into Irreducibles(cid:9) 73
4.1 Historical Background (cid:9) 75
4.2 Trivial Factorizations (cid:9) 76
4.3 Factorization into Irreducibles (cid:9) 79
4.4 Examples of Non-Unique Factorization into Irreducibles(cid:9) 82
4.5 Prime Factorization (cid:9) 86
4.6 Euclidean Domains (cid:9) 90
4.7 Euclidean Quadratic Fields (cid:9) 91
4.8(cid:9) Consequences of Unique Factorization (cid:9) 94
4.9 The Ramanujan-Nagell Theorem (cid:9) 96
4.10 Exercises (cid:9) 99
5 Ideals(cid:9) 101
5.1 Historical Background (cid:9) 102
5.2(cid:9) Prime Factorization of Ideals (cid:9) 105
5.3 The Norm of an Ideal (cid:9) 114
5.4 Nonunique Factorization in Cyclotomic Fields (cid:9) 122
5.5 Exercises (cid:9) 124
II Geometric Methods(cid:9) 127
6 Lattices(cid:9) 129
6.1 Lattices (cid:9) 129
6.2 The Quotient Torus (cid:9) 132
6.3 Exercises (cid:9) 136
7 Minkowsld's Theorem(cid:9) 139
7.1 Minkowski's Theorem (cid:9) 139
7.2 The Two-Squares Theorem (cid:9) 142
7.3 The Four-Squares Theorem (cid:9) 143
7.4 Exercises (cid:9) 144
8 Geometric Representation of Algebraic Numbers(cid:9) 145
8.1(cid:9) The Space mat hb f L' (cid:9) 145
8.2 Exercises (cid:9) 149
Contents(cid:9) ix
9 Class-Group and Class-Number(cid:9) 151
9.1 The Class-Group (cid:9) 152
9.2 An Existence Theorem (cid:9) 153
9.3(cid:9) Finiteness of the Class-Group (cid:9) 157
9.4 How to Make an Ideal Principal (cid:9) 158
9.5(cid:9) Unique Factorization of Elements in an Extension Ring(cid:9) 162
9.6 Exercises (cid:9) 164
III Number-Theoretic Applications(cid:9) 167
10 Computational Methods(cid:9) 169
10.1 Factorization of a Rational Prime (cid:9) 169
10.2 Minkowski's Constants (cid:9) 172
10.3 Some Class-Number Calculations (cid:9) 176
10.4 Tables (cid:9) 179
10.5 Exercises (cid:9) 180
11 Kummer's Special Case of Fermat's Last Theorem(cid:9) 183
11.1 Some History (cid:9) 183
11.2 Elementary Considerations (cid:9) 186
11.3 Kummer's Lemma (cid:9) 189
11.4 Kummer's Theorem (cid:9) 193
11.5 Regular Primes (cid:9) 196
11.6 Exercises (cid:9) 198
12 The Path to the Final Breakthrough(cid:9) 201
12.1 The Wolfskehl Prize (cid:9) 201
12.2 Other Directions (cid:9) 203
12.3 Modular Functions and Elliptic Curves (cid:9) 205
12.4 The Taniyama-Shimura-Weil Conjecture (cid:9) 206
12.5 Frey's Elliptic Equation (cid:9) 207
12.6 The Amateur who Became a Model Professional (cid:9) 207
12.7 Technical Hitch (cid:9) 210
12.8 Flash of Inspiration (cid:9) 211
12.9 Exercises (cid:9) 212
13 Elliptic Curves(cid:9) 213
13.1 Review of Conics (cid:9) 214
13.2 Projective Space (cid:9) 215
13.3 Rational Conics and the Pythagorean Equation (cid:9) 220
13.4 Elliptic Curves (cid:9) 222
13.5 The Tangent/Secant Process (cid:9) 225