Basic Number Theory

ISBN-13: 9783540586555 / Angielski / Miękka / 1995 / 316 str.

Andre Weil

Opis

)tPI(}jlOV, e oxov (10CPUljlr1.'CWV Aiux., llpop. . .dsup.. The first part of this volume is based on a course taught at Princeton University in 1961-62; at that time, an excellent set of notes was prepared by David Cantor, and it was originally my intention to make these notes available to the mathematical public with only quite minor changes. Then, among some old papers of mine, I accidentally came across a long-forgotten manuscript by Chevalley, of pre-war vintage (forgotten, that is to say, both by me and by its author) which, to my taste at least, seemed to have aged very well. It contained a brief but essentially com plete account of the main features of classfield theory, both local and global; and it soon became obvious that the usefulness of the intended volume would be greatly enhanced if I included such a treatment of this topic. It had to be expanded, in accordance with my own plans, but its outline could be preserved without much change. In fact, I have adhered to it rather closely at some critical points."

Recenzje

"L.R. Shafarevich showed me the first edition in autumn 1967 in Moscow and said that this book will be from now on the book about class field theory. In fact it is by far the most complete treatment of the main theorems of algebraic number theory, including function fields over finite constant fields, that appeared in book form. The theory is presented in a uniform way starting with topological fields and Haar measure on related groups, and it includes not only class field theory but also the theory of simple algebras over local and global fields, which serves as a foundation for class field theory. The spirit of the book is the idea that all this is asic number theory' about which elevates the edifice of the theory of automorphic forms and representations and other theories.
To develop this basic number theory on 312 pages efforts a maximum of concentration on the main features. So, there is absolutely no example which illustrates the rather abstract material and brings it nearer to the heart of the reader.
This is not a book for beginners. This book is written in the spirit of the early forties and just this makes it a valuable source of information for everyone who is working about problems related to number and function fields."
Zentralblatt MATH, 823

Spis treści

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.

O autorze

Biography of André Weil

André Weil was born on May 6, 1906 in Paris. After studying mathematics at the École Normale Supérieure and receiving a doctoral degree from the University of Paris in 1928, he held professorial positions in India, France, the United States and Brazil before being appointed to the Institute for Advanced Study, Princeton in 1958, where he remained until he died on August 6, 1998.

André Weil's work laid the foundation for abstract algebraic geometry and the modern theory of abelian varieties. A great deal of his work was directed towards establishing the links between number theory and algebraic geometry and devising modern methods in analytic number theory. Weil was one of the founders, around 1934, of the group that published, under the collective name of  N. Bourbaki, the highly influential multi-volume treatise Eléments de mathématique.