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
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  st ......................................... 145
8.2 Exercises ............................................. 150
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 Format'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-Weil Conjecture .................. 206
12.5 Prey'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 Conies ...................................... 216
13.2 Projective Space ...................................... 217
13.3 Rational Conies and the Pythagorean Equation .......... 222
13.4 Elliptic Curves ....................................... 224
13.5 The Tangent/Secant Process ............................ 227
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 Prey Elliptic Curve ............................... 259
15.2 The Taniyama-Shimura-Weil Conjectme ................... 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.l Quadratic Equations in  m .............................. 280
A.2 The Units of m ........................................ 282
A.3 Quadratic Residues ..................................... 287
A.4 Exercises .............................................. 296
В Dirichlet's Units Theorem .................................. 299
B.l 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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