代数数论
2003-11
世界图书出版公司(此信息作废)
Serge Lang
357
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The present book gives an exposition of the classical basic algebraic and analytic number theory and supersedes my Algebraic Numbers,including much more material, e.g. the class field theory on which I make further comments at the appropriate place later. For different points of view, the reader is encouraged to read the collection of papers from the Brighton Symposium (edited by Cassels-Frohlich),the Artin-Tate notes on class field theory, Well's book on Basic Number Theory, Borevich-Shafarevich's Number Theory, and also older books like those of Weber, Hasse, Hecke, and HUbert's Zahlbericht. It seems that over the years, everything that has been done has proved useful, theoretically or as examples, for the further development of the theory. Old,and seemingly isolated special cases have continuously acquired renewedsignificance, often after half a century or more.
Part One General Basic Theory CHAPTER IAlgebraic Integers 1. Localization 2. Integral closure 3. Prime ideals 4. Chinese remainder theorem 5. Galois extensions 6. Dedekind rings 7. Discrete valuation rings 8. Explicit faetorization of a prime 9. Projective modules over Dedekind rings CHAPTER II Completions 1. Definitions and completions 2. Polynomials in complete fields 3. Some filtrations 4. Unramified extensions 5. Tamely ramified extensions CHAPTER IV Cyclotomic Fields 1. Roots of unity 2. Quadratic fields 3. Gauss sums 4. Relations in ideal classes CHARTER V Parallelotopes 1. The product formula 2. Lattice points in parallelotopes 3. A volume computation 4. Minkowski's constant CHAPTER VI The Ideal Function 1. Generalized ideal classes 2. Lattice points in homogeneously expanding domains 3. The number of ideals in a given class CHAPTER VII Ideles and Adeles 1. Restricted direct products 2. Adeles 3. Ideles 4. Generalized ideal class groups; relations with idele classes 5. Embedding of k* in the idele classes 6. Galois operation on ideles and idele classes CHAPTER VIII Elementary Properties of the Zeta Function and L-series 1. Lemmas on Dirichlet series 2. Zeta function of a number field 3. The L-series 4. Density of primes in arithmetic progressions 5. FaRings' finiteness theoremPart Two Class Field TheoryPart Three Analytic TheoryBibliographyIndex
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