On the basis of the number of

by André Weil

Basic Number Theory (English Edition Published in China) by Andre Weil The first part of this volume is based on a course taught at PrincetonUniversity in 1961-62; at that time, an excellent set of notes was preparedby David Cantor, and it was originally my intention to make these notesavailable to the mathematical public with only quite minor changes.Then, among some old papers of mine, I accidentally came across along=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 andglobal; and it soon became obvious that the usefulness of the intendedvolume would be greatly enhanced if I included such a treatment of thistopic. It had to be expanded, in accordance with my own plans, but itsoutline could be preserved without much change. In fact, I have adheredto it rather closely at some critical points. To improve upon Hecke, in a treatment along classical lines of thetheory of algebrai~ numbers, would be a futile and impossible task. Aswill become apparent from the first pages of this book, I have rathertried to draw the conclusions from the developments of the last thirtyyears, whereby locally compact groups, measure and integration havebeen seen to play an increasingly important role in classical number-theory. I

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Published January 1, 2010