Basic number theory

Includes index

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"

The first part of this volume is based on a course taught at Princeton University in 1961-62. The author came upon a long-forgotten manuscript, which contained a brief but essentially complete account of the main features of classfield theory, both local and global, the inclusion of which greatly enhanced this volume. The author has tried to draw conclusions from the developments of the last thirty years, whereby locally compact groups, measure and integration have been seen to play an increasingly important role in classical number theory.