| 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 theorem Part Two Class Field Theory Part Three Analytic Theory Bibliography Index |
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