Algebraic number theory and Fermat’s last theorem

338 Pages·2016·2.426 MB·English

by  Stewart, Ian, Tall, David Orme

Mathematics
FOURTH
EDITION
A N T
Updated to reflect current research, Algebraic Number Theory and
A lgebrAic umber heory
Fermat’s Last Theorem, Fourth Edition introduces fundamental ideas of l
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algebraic numbers and explores one of the most intriguing stories in the e
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history of mathematics—the quest for a proof of Fermat’s Last Theorem. r
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The authors use this celebrated theorem to motivate a general study of ic F l T

the theory of algebraic numbers from a relatively concrete point of view. N ANd ermAT’s AsT heorem
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Readers will see how Wiles’s proof of Fermat’s Last Theorem opened many m
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new areas for future work.
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New to the Fourth Edition T n FOURnTH EDITION n
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• Provides up-to-date information on unique prime factorization for e
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real quadratic number fields, especially Harper’s proof that Z(√14) is r
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Euclidean A
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• Presents an important new result: Miha˘ilescu’s proof of the Catalan d

conjecture of 1844 F
e xn + yn = zn
• Revises and expands one chapter into two, covering classical ideas r
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about modular functions and highlighting the new ideas of Frey, Wiles, A
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and others that led to the long-sought proof of Fermat’s Last Theorem ’
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• Improves and updates the index, figures, bibliography, further reading l
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list, and historical remarks s
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Written by preeminent mathematicians Ian Stewart and David Tall, this text h
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continues to teach readers how to extend properties of natural numbers to o
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more general number structures, including algebraic number fields and their e
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rings of algebraic integers. It also explains how basic notions from the theory
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of algebraic numbers can be used to solve problems in number theory.
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FOURTH EDITION
Ian Stewart
University of Warwick
United Kingdom

David Tall
University of Warwick
United Kingdom
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2016 by Taylor & Francis Group, LLC
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Version Date: 20150616
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Contents
Preface to the Third Edition ix
Preface to the Fourth Edition xv
Index of Notation xvii
The Origins of Algebraic Number Theory 1
I Algebraic Methods 9
1 Algebraic Background 11
1.1 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Factorization of Polynomials. . . . . . . . . . . . . . . . . . 15
1.3 Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4 Symmetric Polynomials . . . . . . . . . . . . . . . . . . . . 24
1.5 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6 Free Abelian Groups . . . . . . . . . . . . . . . . . . . . . . 28
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Algebraic Numbers 37
2.1 Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Conjugates and Discriminants . . . . . . . . . . . . . . . . . 40
2.3 Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Integral Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Norms and Traces . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 Rings of Integers . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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vi Contents
3 Quadratic and Cyclotomic Fields 63
3.1 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Cyclotomic Fields. . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Factorization into Irreducibles 75
4.1 Historical Background . . . . . . . . . . . . . . . . . . . . . 77
4.2 Trivial Factorizations . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Factorization into Irreducibles . . . . . . . . . . . . . . . . . 81
4.4 Examples of Non-Unique Factorization into Irreducibles . . 84
4.5 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . 88
4.6 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . 92
4.7 Euclidean Quadratic Fields . . . . . . . . . . . . . . . . . . 93
4.8 Consequences of Unique Factorization . . . . . . . . . . . . 96
4.9 The Ramanujan-Nagell Theorem . . . . . . . . . . . . . . . 98
4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Ideals 103
5.1 Historical Background . . . . . . . . . . . . . . . . . . . . . 104
5.2 Prime Factorization of Ideals . . . . . . . . . . . . . . . . . 107
5.3 The Norm of an Ideal . . . . . . . . . . . . . . . . . . . . . 116
5.4 Non-Unique Factorization in Cyclotomic Fields . . . . . . . 124
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
II Geometric Methods 129
6 Lattices 131
6.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 The Quotient Torus . . . . . . . . . . . . . . . . . . . . . . 134
6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7 Minkowski’s Theorem 139
7.1 Minkowski’s Theorem . . . . . . . . . . . . . . . . . . . . . 139
7.2 The Two-Squares Theorem . . . . . . . . . . . . . . . . . . 142
7.3 The Four-Squares Theorem . . . . . . . . . . . . . . . . . . 143
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8 Geometric Representation of Algebraic Numbers 145
8.1 The Space Lst . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
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Contents vii
9 Class-Group and Class-Number 151
9.1 The Class-Group . . . . . . . . . . . . . . . . . . . . . . . . 152
9.2 An Existence Theorem . . . . . . . . . . . . . . . . . . . . . 153
9.3 Finiteness of the Class-Group . . . . . . . . . . . . . . . . . 157
9.4 How to Make an Ideal Principal . . . . . . . . . . . . . . . . 158
9.5 Unique Factorization of Elements in an Extension Ring . . 162
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
III Number-Theoretic Applications 167
10 Computational Methods 169
10.1 Factorization of a Rational Prime . . . . . . . . . . . . . . . 169
10.2 Minkowski Constants . . . . . . . . . . . . . . . . . . . . . . 172
10.3 Some Class-Number Calculations . . . . . . . . . . . . . . . 176
10.4 Table of Class-Numbers . . . . . . . . . . . . . . . . . . . . 179
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
11 Kummer’s Special Case of Fermat’s Last Theorem 183
11.1 Some History . . . . . . . . . . . . . . . . . . . . . . . . . . 183
11.2 Elementary Considerations . . . . . . . . . . . . . . . . . . 186
11.3 Kummer’s Lemma . . . . . . . . . . . . . . . . . . . . . . . 188
11.4 Kummer’s Theorem . . . . . . . . . . . . . . . . . . . . . . 193
11.5 Regular Primes . . . . . . . . . . . . . . . . . . . . . . . . . 196
11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12 The Path to the Final Breakthrough 201
12.1 The Wolfskehl Prize . . . . . . . . . . . . . . . . . . . . . . 201
12.2 Other Directions . . . . . . . . . . . . . . . . . . . . . . . . 203
12.3 Modular Functions and Elliptic Curves . . . . . . . . . . . . 205
12.4 The Taniyama–Shimura–WeilConjecture . . . . . . . . . . 206
12.5 Frey’s Elliptic Equation . . . . . . . . . . . . . . . . . . . . 207
12.6 The Amateur who Became a Model Professional . . . . . . 208
12.7 Technical Hitch . . . . . . . . . . . . . . . . . . . . . . . . . 211
12.8 Flash of Inspiration. . . . . . . . . . . . . . . . . . . . . . . 211
12.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
13 Elliptic Curves 215
13.1 Review of Conics . . . . . . . . . . . . . . . . . . . . . . . . 216
13.2 Projective Space . . . . . . . . . . . . . . . . . . . . . . . . 217
13.3 Rational Conics and the PythagoreanEquation . . . . . . . 222
13.4 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . 224
13.5 The Tangent/Secant Process . . . . . . . . . . . . . . . . . 227
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13.6 Group Structure on an Elliptic Curve . . . . . . . . . . . . 228
13.7 Applications to Diophantine Equations . . . . . . . . . . . . 232
13.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
14 Elliptic Functions 235
14.1 Trigonometry Meets Diophantus . . . . . . . . . . . . . . . 235
14.2 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . 243
14.3 Legendre and Weierstrass . . . . . . . . . . . . . . . . . . . 249
14.4 Modular Functions . . . . . . . . . . . . . . . . . . . . . . . 251
14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
15 Wiles’s Strategy and Recent Developments 259
15.1 The Frey Elliptic Curve . . . . . . . . . . . . . . . . . . . . 259
15.2 The Taniyama–Shimura–WeilConjecture . . . . . . . . . . 261
15.3 Sketch Proof of Fermat’s Last Theorem . . . . . . . . . . . 264
15.4 Recent Developments. . . . . . . . . . . . . . . . . . . . . . 266
15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
IV Appendices 277
A Quadratic Residues 279
A.1 Quadratic Equations in Zm . . . . . . . . . . . . . . . . . . 280
A.2 The Units of Zm . . . . . . . . . . . . . . . . . . . . . . . . 282
A.3 Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . 287
A.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
B Dirichlet’s Units Theorem 299
B.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 299
B.2 Logarithmic Space . . . . . . . . . . . . . . . . . . . . . . . 300
B.3 Embedding the Unit Group in Logarithmic Space . . . . . . 301
B.4 Dirichlet’s Theorem . . . . . . . . . . . . . . . . . . . . . . 302
B.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Bibliography 309
Index 317
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Preface to the Third Edition
Thetitleofthisbookindicatesadualpurpose. Ourfirstaimistointroduce
fundamental ideas of algebraic numbers. The second is to tell one of the
mostintriguingstoriesinthehistoryofmathematics—thequestforaproof
of Fermat’s Last Theorem. We use this celebrated theorem to motivate
a general study of the theory of algebraic numbers, from a reasonably
concretepointofview. Therangeoftopicsthatwecoverisselectedtoallow
students to make early progress in understanding the necessary concepts.
‘Algebraic Number Theory’ can be read in two distinct ways. One is
the theory of numbers viewed algebraically, the other is the study of al-
gebraic numbers. Both apply here. We illustrate how basic notions from
the theory of algebraic numbers may be used to solve problems in number
theory. However, our main focus is to extend properties of the natural
numbers to more general number structures: algebraic number fields, and
their rings of algebraic integers. These structures have most of the stan-
dard properties that we associate with ordinary whole numbers, but some
subtle properties concerning primes and factorization sometimes fail to
generalize.
ADiophantineequation(namedafterDiophantusofAlexandria,who—
it is thought—lived around 250 and whose book Arithmetica systematized
such concepts) is a polynomial equation, or a system of polynomial equa-
tions, that is to be solved in integers or rational numbers. The central
problem of this book concerns solutions of a very special Diophantine
equation:
xn+yn =zn
wheretheexponentnisapositiveinteger. Forn=2therearemanyinteger
solutions—in fact, infinitely many—which neatly relate to the theorem of
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