)tPI(}jlOV, eõxov (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 comยญ 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
CONTENT
I. Elementary Theory -- I. Locally compact fields -- II. Lattices and duality over local fields -- III. Places of A-fields -- IV. Adeles -- V. Algebraic number-fields -- VI. The theorem of Riemann-Roch -- VII. Zeta-functions of A-fields -- VIII. Traces and norms -- II. Classfield Theory -- IX. Simple algebras -- X. Simple algebras over local fields -- XI. Simple algebras over A-fields -- XII. Local classfield theory -- XIII. Global classfield theory -- Notes to the text -- Appendix I. The transfer theorem -- Appendix III. Shafarevitchโ{128}{153}s theorem -- Appendix IV. The Herbrand distribution -- Index of definitions
