Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften (322))

"The present book has as its aim to resolve a discrepancy in the textbook literature and ... to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry. ... Despite this exacting program, the book remains an introduction to algebraic number theory for the beginner... The author discusses the classical concepts from the viewpoint of Arakelov theory.... The treatment of class field theory is ... particularly rich in illustrating complements, hints for further study, and concrete examples.... The concluding chapter VII on zeta-functions and L-series is another outstanding advantage of the present textbook.... The book is, without any doubt, the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available." W. Kleinert in Z.blatt f. Math., 1992 "The author's enthusiasm for this topic is rarely as evident for the reader as in this book. - A good book, a beautiful book." F. Lorenz in Jber. DMV 1995 "The present work is written in a very careful and masterly fashion. It does not show the pains that it must have caused even an expert like Neukirch. It undoubtedly is liable to become a classic; the more so as recent developments have been taken into account which will not be outdated quickly. Not only must it be missing from the library of no number theorist, but it can simply be recommended to every mathematician who wants to get an idea of modern arithmetic." J. Schoissengeier in Montatshefte Mathematik 1994 Contents Chapter I: Algebraic Integers 1. The Gaussian Integers 2. Integrality 3. Ideals 4. Lattices 5. Minkowski Theory 6. The Class Number 7. Dirichlet's Unit Theorem 8. Extensions of Dedekind Domains 9. Hilbert's Ramification Theory 10. Cyclotomic Fields 11. Localization 12. Orders 13. One-dimensional Schemes 14. Function Fields Chapter II: The Theory of Valuations 1. The p-adic Numbers 2. The p-adic Absolute Value 3. Valuations 4. Completions 5. Local Fields 6. Henselian Fields 7. Unramified and Tamely Ramified Extensions 8. Extensions of Valuations 9. Galois Theory of Valuations 10. Higher Ramification Groups Chapter III: Riemann-Roch Theory 1. Primes 2. Different and Discriminant 3. Riemann-Roch 4. Metrized o-Modules 5. Grothendieck Groups 6. The Chern Character 7. Grothendieck-Riemann-Roch 8. The Euler-Minkowski Characteristic Chapter IV: Abstract Class Field Theory 1. Infinite Galois Theory 2. Projective and Inductive Limits 3. Abstract Galois Theory 4. Abstract Valuation Theory 5. The Reciprocity Map 6. The General Reciprocity Law 7. The Herbrand Quotient Chapter V: Local Class Field Theory 1. The Local Reciprocity Law 2. The Norm Residue Symbol over Qp 3. The Hilbert Symbol 4. Formal Groups 5. Generalized Cyclotomic Theory 6. Higher Ramification Groups Chapter VI: Global Class Field Theory l. Ideles and Idele Classes 2. Ideles in Field Extensions 3. The Herbrand Quotient of the Idele Class Group 4. The Class Field Axiom 5. The Global Reciprocity Law 6. Global Class Fields 7. The Ideal-Theoretic Version of Class Field Theory 8. The Reciprocity Law of the Power Residues Chapter VII: Zeta Functions and L-series 1. The Riemann Zeta Function 2. Dirichlet L-series 3. Theta Series 4. The Higher-dimensional Gamma Function 5. The Dedekind Zeta Function 6. Hecke Characters 7. Theta Series of Algebraic Number Fields 8. Hecke L-series 9. Values of Dirichlet L-series at Integer Points 10. Artin L-series 11. The Artin Conductor 12. The Functional Equation of Artin L-series 13. Density Theorems Bibliography Index

This introduction to algebraic number theory discusses the classical concepts from the viewpoint of Arakelov theory. The treatment of class theory is particularly rich in illustrating complements, offering hints for further study, and providing concrete examples. It is the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available.