Preface .......................................... I
Suggested Course Outlines............. xiii
Chapter 1: Algebraic Numbers.................. 1
1.1 Origin and Foundations............. 11.2 Algebraic Numbers and Number Fields.............................. 13
1.3 Discriminants, Norms and Traces...................................... 25
1.4 Algebraic Integers and Integral Bases..................................... 32
1.5 Factorization and Divisibility................ 48
1.6 Applications of Unique Factorization............... 53
1.7 Applications: Factoring in Z Using Cubic Integers.............67
Chapter 2: Arithmetic of Number Fields................................ 73
2.1 Quadratic Fields........................................ 73
2.2 Cyclotomic Fields................................ 81
2.3 Units in Number Rings............ 89
2.4 Geometry of Numbers..................... 93
2.5 Dirichlet's Unit Theorem........... 108
2.6 Application: The Number Field Sieve...................... 117
Chapter 3: Ideal Theory............................ 127
3.1 Properties of Ideals............................................. 127
3.2 PID's and UFD's......................................... 142
3.3 Norms of Ideals...........................................148
3.4. Ideal Classes-The Class Group............. 153
3.5 Class Numbers of Quadratic Fields................. 159
3.6 Cyclotomic Fields and Kummer's Theorem-
Bernoulli Numbers and Irregular Primes................. 170
3.7 Cryptography in Quadratic Fields................. 183
Chapter 4: Ideal Decomposition in Extension Fields......................... 193
4.1 Inertia, Ramification and Splitting..................... 193
4.2 The Different and Discriminant.................. 209
4.3 Galois Theory and Decomposition.................. 232
4.4 The Kronecker-Weber Theorem............ 256
4.5 An Application-Primality Testing.................. 264
Chapter 5: Reciprocity Laws........................ 273
5.1 Cubic Reciprocity........................ 273
5.2 The Biquadratic Reciprocity Law...................... 289
5.3 The Stickelberger Relation...................... 306
5.4 The Eisenstein Reciprocity Law................. 325
5.5 Elliptic Curves, Factoring and Primality.................. 333
Appendix A: Abstract Algebra............................... 352
Appendix B: Sequences and Series.... 383
Appendix C: Galois Theory................. 393
Appendix D: The Greek Alphabet................ 402
Appendix E: Latin Phrases....... 403
Solutions to Odd Numbered Exercises................... 405
Bibliography.......................................... 459
List of Symbols....................................... 464
Index................................................. 466
About the Author................................................. 483
Last updated: September 10, 2003
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