Basic Number Theory (Paperback, 3)

I. Elementary Theory.- I. Locally compact fields.- 1. Finite fields.- 2. The module in a locally compact field.- 3. Classification of locally compact fields.- 4. Structure of p-fields.- II. Lattices and duality over local fields.- 1. Norms.- 2. Lattices.- 3. Multiplicative structure of local fields.- 4. Lattices over R.- 5. Duality over local fields.- III. Places of A-fields.- 1. A-fields and their completions.- 2. Tensor-products of commutative fields.- 3. Traces and norms.- 4. Tensor-products of A-fields and local fields.- IV. Adeles.- 1. Adeles of A-fields.- 2. The main theorems.- 3. Ideles.- 4. Ideles of A-fields.- V. Algebraic number-fields.- 1. Orders in algebras over Q.- 2. Lattices over algebraic number-fields.- 3. Ideals.- 4. Fundamental sets.- VI. The theorem of Riemann-Roch.- VII. Zeta-functions of A-fields.- 1. Convergence of Euler products.- 2. Fourier transforms and standard functions.- 3. Quasicharacters.- 4. Quasicharacters of A-fields.- 5. The functional equation.- 6. The Dedekind zeta-function.- 7. L-functions.- 8. The coefficients of the L-series.- VIII. Traces and norms.- 1. Traces and norms in local fields.- 2. Calculation of the different.- 3. Ramification theory.- 4. Traces and norms in A-fields.- 5. Splitting places in separable extensions.- 6. An application to inseparable extensions.- II. Classfield Theory.- IX. Simple algebras.- 1. Structure of simple algebras.- 2. The representations of a simple algebra.- 3. Factor-sets and the Brauer group.- 4. Cyclic factor-sets.- 5. Special cyclic factor-sets.- X. Simple algebras over local fields.- 1. Orders and lattices.- 2. Traces and norms.- 3. Computation of some integrals.- XI. Simple algebras over A-fields.- 1. Ramification.- 2. The zeta-function of a simple algebra.- 3. Norms in simple algebras.- 4. Simple algebras over algebraic number-fields.- XII. Local classfield theory.- 1. The formalism of classfield theory.- 2. The Brauer group of a local field.- 3. The canonical morphism.- 4. Ramification of abelian extensions.- 5. The transfer.- XIII. Global classfield theory.- 1. The canonical pairing.- 2. An elementary lemma.- 3. Hasse's "law of reciprocity".- 4. Classfield theory for Q.- 5. The Hilbert symbol.- 6. The Brauer group of an A-field.- 7. The Hilbert p-symbol.- 8. The kernel of the canonical morphism.- 9. The main theorems.- 10. Local behavior of abelian extensions.- 11. "Classical" classfield theory.- 12. "Coronidis loco".- Notes to the text.- Appendix I. The transfer theorem.- Appendix III. Shafarevitch's theorem.- Appendix IV. The Herbrand distribution.- Index of definitions.