Basic Number Theory

The first part of this volume is based on a course taught at Princeton University in 1961-62; at that time, an excellent set ofnotes 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 ChevaIley, 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 weIl. It contained abrief but essentially com- plete account of the main features of c1assfield theory, both local and global; and it soon became obvious that the usefulness of the intended volume would be greatly enhanced if I 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 to it rather c10sely at some critical points.
To improve upon Hecke, in a treatment along classicallines 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 number- theory. In the days of Dirichlet and Hermite, and even of Minkowski, the appeal to "continuous variables" in arithmetical questions may weIl 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 completions of the prime field, one which is neither more nor less interesting to the arithmetician than its p-adic companions, and that there is at least one language and one technique, that of the adeles, for bringing themall together under one roof and making them cooperate for a common purpose. It is needless here to go into the history of these developments; suffice it to mention such names as Hensel, Hasse, ChevaIley, Artin; every one of these, and more recently Iwasawa, Tate, Tamagawa, helped to make some significant step forward along this road. Once the presence of the real field, albeit at infinite distance, ceases to be regarded as a necessary ingredient in the arithmetician's brew, itgoes without saying that the function-fields over finite fields must be granted a fully simultaneous treatment with number-fields, instead of the segregated status, and at best the separate but equal facilities, which hitherto have been their lot. That, far from losing by such treatment, both races stand to gain by it, is one fact which will, I hope, clearly emerge from this book.