Table of Contents 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 §l. Primes 183 §2. Different and Discriminant 194 §3. Riemann-Roch 208 §4. Metrized a-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 §l. The Local Reciprocity Law 317 §2. The Norm Residue Symbol over Qp 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 §I. Ideles and Idele Classes 357 §2. Ideles in Field Extensions 368 §3. The Herbrand Quotient of the Idele 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
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