Weil A. Basic Number Theory 1995

Textbook in PDF format

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.
Elementary Theory
Locally compact fieldsLattices and duality over local fields
Places of A•fields
Adeles
Algebraic number-fields
The theorem of Riemann-Roch
Zeta-functions of A-fields
Traces and norms
Classfield Theory
Simple algebras
Simple algebras over local fields
Simple algebras over A-fields
Local classfield theory
Global classfield theory