Preface ..................................................... ix
About the Author .......................................... xiii
Suggested Course Outlines ................................... xv
1 Integral Domains, Ideals, and Unique Factorization ........... 1
1.1 Integral Domains ........................................ 1
1.2 Factorization Domains ................................... 7
1.3 Ideals ................................................. 15
1.4 Noetherian and Principal Ideal Domains ................. 20
1.5 Dedekind Domains ....................................... 25
1.6 Algebraic Numbers and Number Fields .................... 35
1.7 Quadratic Fields ....................................... 44
2 Field Extensions ............................................ 55
2.1 Automorphisms, Fixed Points, and Galois
Groups ................................................. 55
2.2 Norms and Traces ....................................... 65
2.3 Integral Bases and Discriminants ....................... 70
2.4 Norms of Ideals ........................................ 83
3 Class Groups ................................................ 87
3.1 Binary Quadratic Forms ................................. 87
3.2 Forms and Ideals ....................................... 96
3.3 Geometry of Numbers and the Ideal Class Group ......... 108
3.4 Units in Number Rings ................................. 122
3.5 Dirichlet's Unit Theorem .............................. 130
4 Applications: Equations and Sieves ......................... 139
4.1 Prime Power Representation ............................ 139
4.2 Bachet's Equation ..................................... 145
4.3 The Fermat Equation ................................... 149
4.4 Factoring ............................................. 165
4.5 The Number Field Sieve ................................ 174
5 Ideal Decomposition in Number Fields ....................... 181
5.1 Inertia, Ramification, and Splitting of Prime
Ideals ................................................ 181
5.2 The Different and Discriminant ........................ 196
5.3 Ramification .......................................... 213
5.4 Galois Theory and Decomposition ....................... 221
5.5 Kummer Extensions and Class-Field Theory .............. 233
5.6 The Kronecker-Weber Theorem ........................... 244
5.7 An Application-Primality Testing ...................... 255
6 Reciprocity Laws ........................................... 261
6.1 Cubic Reciprocity ..................................... 261
6.2 The Biquadratic Reciprocity Law ....................... 278
6.3 The Stickelberger Relation ............................ 294
6.4 The Eisenstein Reciprocity Law ........................ 311
Appendix A: Abstract Algebra .................................. 319
Appendix B: Sequences and Series .............................. 345
Appendix C: The Greek Alphabet ................................ 355
Appendix D: Latin Phrases ..................................... 357
Bibliography .................................................. 359
Solutions to Odd-Numbered Exercises ........................... 365
Index ......................................................... 407
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