Algebraic Number Theory

by Neukirch, Jurgen; Schappacher, Norbert

• ISBN13: 9783540653998
• ISBN10: 3540653996
• Format: Hardcover
• Copyright: 1999-05-01
• Publisher: Springer Nature

Summary

From the review: "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: Zentralblatt für Mathematik, 1992

Table of Contents

Chapter I: Algebraic Integers	1	(98)
1. The Gaussian Integers	1	(4)
2. Integrality	5	(11)
3. Ideals	16	(7)
4. Lattices	23	(5)
5. Minkowski Theory	28	(6)
6. The Class Number	34	(5)
7. Dirichlet's Unit Theorem	39	(5)
8. Extensions of Dedekind Domains	44	(9)
9. Hilbert's Ramification Theory	53	(5)
10. Cyclotomic Fields	58	(7)
11. Localization	65	(7)
12. Orders	72	(12)
13. One-dimensional Schemes	84	(10)
14. Function Fields	94	(5)
Chapter II: The Theory of Valuations	99	(84)
1. The p-adic Numbers	99	(7)
2. The p-adic Absolute Value	106	(10)
3. Valuations	116	(7)
4. Completions	123	(11)
5. Local Fields	134	(9)
6. Henselian Fields	143	(9)
7. Unramified and Tamely Ramified Extensions	152	(8)
8. Extensions of Valuations	160	(6)
9. Galois Theory of Valuations	166	(10)
10. Higher Ramification Groups	176	(7)
Chapter III: Riemann-Roch Theory	183	(78)
1. Primes	183	(11)
2. Different and Discriminant	194	(14)
3. Riemann-Roch	208	(16)
4. Metrized o-Modules	224	(9)
5. Grothendieck Groups	233	(10)
6. The Chern Character	243	(3)
7. Grothendieck-Riemann-Roch	246	(9)
8. The Euler-Minkowski Characteristic	255	(6)
Chapter IV: Abstract Class Field Theory	261	(56)
1. Infinite Galois Theory	261	(4)
2. Projective and Inductive Limits	265	(10)
3. Abstract Galois Theory	275	(9)
4. Abstract Valuation Theory	284	(6)
5. The Reciprocity Map	290	(9)
6. The General Reciprocity Law	299	(11)
7. The Herbrand Quotient	310	(7)
Chapter V: Local Class Field Theory	317	(40)
1. The Local Reciprocity Law	317	(10)
2. The Norm Residue Symbol over Q(p)	327	(6)
3. The Hilbert Symbol	333	(8)
4. Formal Groups	341	(5)
5. Generalized Cyclotomic Theory	346	(6)
6. Higher Ramification Groups	352	(5)
Chapter VI: Global Class Field Theory	357	(62)
1. Ideles and Idele Classes	357	(11)
2. Ideles in Field Extensions	368	(5)
3. The Herbrand Quotient of the Idele Class Group	373	(7)
4. The Class Field Axiom	380	(5)
5. The Global Reciprocity Law	385	(10)
6. Global Class Fields	395	(10)
7. The Ideal-Theoretic Version of Class Field Theory	405	(9)
8. The Reciprocity Law of the Power Residues	414	(5)
Chapter VII: Zeta Functions and L-series	419	(132)
1. The Riemann Zeta Function	419	(15)
2. Dirichlet L-series	434	(9)
3. Theta Series	443	(10)
4. The Higher-dimensional Gamma Function	453	(4)
5. The Dedekind Zeta Function	457	(13)
6. Hecke Characters	470	(14)
7. Theta Series of Algebraic Number Fields	484	(9)
8. Hecke L-series	493	(11)
9. Values of Dirichlet L-series at Integer Points	504	(13)
10. Artin L-series	517	(10)
11. The Artin Conductor	527	(8)
12. The Functional Equation of Artin L-series	535	(7)
13. Density Theorems	542	(9)
Bibliography	551	(8)
Index	559