Basic Number Theory
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der Anwendungsgebiete Band 144
Herausgegeben von
J.
L. Doob . E. Heinz' Heinz . F. Hirzebruch . E. Hopf' H. Hopf
Lane . W. Magnus' Magnus . D. Mumford W. Maak . S. Mac Lane' Schmidt . D. S. Scott· Scott . K. Stein M. M. Postnikov . F. K. Schmidt·
C;eschä~t~ührende C;escha~t~uhrende
Herausgeber
B. Eckmann und B. L. van der Waerden
Andre Weil
Basic Number Theory
Springer-Verlag Berlin Heidelberg New York Y ork 1967
Professor Andre Wei1 The Institute for Advanced Study, Princeton, N. J. 08540 Geschăftsfiihrende
Herausgeber:
Prof. Dr. B. Eckmann
Eidgenossische Technische Hochschule Ziirich
Prof. Dr. B. L. van der Waerden
Mathematisches Institut der Universităt Ziirich
ISBN 978-3-662-00048-9 ISBN 978-3-662-00046-5 (eBook) DOI 10.1007/978-3-662-00046-5
AII rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm andJor microcard) or by other procedure without written permis sion from Springer-Verlag © by Springer-Verlag Berlin' Heidelberg 1967 Library ofCongress Catalog Card Number 67-25021
Softcover reprint ofthe hardcover Ist edition 1967
Title-No. 5127
Foreword }tPZ()jlOV, g~oxov UO((JUijlrXr:WV
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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 ofnotes 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 ChevaIley, of pre-war vintage (forgotten, long-forgotten manuscript by Chevalley, that is to say, both by me and by its author) which, to my taste at least, weIl. It contained aabrief brief but essentially comseemed to have aged very well. plete account of the main features of c1assfield class field theory, both local and global; and it soon became obvious that the usefulness of the intended volume would be greatly enhanced if I included inc1uded 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 c10sely at some critical points. to it rather closely classicallines To improve upon Hecke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task. As will become apparent from the first pages of this book, I have rather tried to draw the 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 numbertheory. In the days of Dirichlet and Hermite, and even of Minkowski, weIl the appeal to "continuous variables" in arithmetical questions may well have seemed to come out of some magician's bag of tricks. In retrospect, we see now that the real numbers appear there as one of the infinitely many c
