ISBN: 3-540-65399-6
TITLE: Algebraic Number Theory
AUTHOR: Neukirch, Jrgen
TOC:

Chapter I: Algebraic Integers 1
 1. The Gaussian Integers 1
 2. Integrality 5
 3. Ideals 16
 4. Lattices 23
 5. Minkowski Theory 28
 6. The Class Number 34
 7. Dirichlet's Unit Theorem 39
 8. Extensions of Dedekind Domains 44
 9. Hilbert's Ramification Theory 53
 10. Cyclotomic Fields 58
 11. Localization 65
 12. Orders 72
 13. One-dimensional Schemes 84
 14. Function Fields 94
Chapter II: The Theory of Valuations 99
 1. The p-adic Numbers 99
 2. The p-adic Absolute Value 106
 3. Valuations 116
 4. Completions 123
 5. Local Fields 134
 6. Henselian Fields 143
 7. Unramified and Tamely Ramified Extensions 152
 8. Extensions of Valuations 160
 9. Galois Theory of Valuations 166
 10. Higher Ramification Groups 176
Chapter III: Riemann-Roch Theory 183
 1. Primes 183
 2. Different and Discriminant 194
 3. Riemann-Roch 208
 4. Metrized O-Modules 224
 5. Grothendieck Groups 233
 6. The Chern Character 243
 7. Grothendieck-Riemann-Roch 246
 8. The Euler-Minkowski Characteristic 255
Chapter IV: Abstract Class Field Theory 261
 1. Infinite Galois Theory 261
 2. Projective and Inductive Limits 265
 3. Abstract Galois Theory 275
 4. Abstract Valuation Theory 284
 5. The Reciprocity Map 290
 6. The General Reciprocity Law 299
 7. The Herbrand Quotient 310
Chapter V: Local Class Field Theory 317
 1. The Local Reciprocity Law 317
 2. The Norm Residue Symbol over Q_p 327
 3. The Hilbert Symbol 333
 4. Formal Groups 341
 5. Generalized Cyclotomic Theory 346
 6. Higher Ramification Groups 352
Chapter VI: Global Class Field Theory 357
 1. Idles and Idle Classes 357
 2. Idles in Field Extensions 368
 3. The Herbrand Quotient of the Idle Class Group 373
 4. The Class Field Axiom 380
 5. The Global Reciprocity Law 385
 6. Global Class Fields 395
 7. The Ideal-Theoretic Version of Class Field Theory 405
 8. The Reciprocity Law of the Power Residues 414
Chapter VII: Zeta Functions and L-series 419
 1. The Riemann Zeta Function 419
 2. Dirichlet L-series 434
 3. Theta Series 443
 4. The Higher-dimensional Gamma Function 453
 5. The Dedekind Zeta Function 457
 6. Hecke Characters 470
 7. Theta Series of Algebraic Number Fields 484
 8. Hecke L-series 493
 9. Values of Dirichlet L-series at Integer Points 504
 10. Artin L-series 517
 11. The Artin Conductor 527
 12. The Functional Equation of Artin L-series 535
 13. Density Theorems 542
Bibliography 551
Index 559
END
