Inhalt

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#x201D.- § 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".- Index of definitions.