基础数论(英文版)

　　本书是一部学习“类域论”的非常好的教材。学习本书不需要任何数论的基础知识，但需要熟知局部紧Abel环，Pontryagin对偶性以及群上的Haar测度的标准定理。此外，本书不适于代数数论的初学者使用。
　　

chronological table
prerequisites and notations
table of notations
part i. elementary theory
chapter i. locally compact fields
1. finite fields
2. the module in a locally compact field
3. classification of locally compact fields
4. structure of p-fields
chapter ii. lattices and duality over local fields
1. norms
2. lattices
3. multiplicative structure of local fields
4. lattices over r
5. duality over local fields
chapter iii. places of a-fields
1. a-fields and their completions
2. tensor-products of commutative fields
3. traces and norms
4. tensor-products of a-fields and local fields

　　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 algebraic 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. In the days of Dirichlet and Hermite, and even of Minkowski,the appeal to "continuous variables" in arithmetical questions may wellhave 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 infinitelymany completions of the prime field, one which is neither more nor lessinteresting to the arithmetician than its p-adic companions, and thatthere is at least one language and one technique, that of the adeles, forbringing them all together under one roof and making them cooperatefor a common purpose. It is needless here to go into the history of thesedevelopments; suffice it to mention such names as Hensel, Hasse,Chevalley, Artin; every one of these, and more recently Iwasawa, Tate,Tamagawa, helped to make some significant step forward along thisroad. Once the presence of the real field, albeit at infinite distance, ceasesto be regarded as a necessary ingredient in the arithmetician's brew, itgoes without saying that the function-fields over finite fields must begranted a fully simultaneous treatment with number-fields, instead ofthe segregated status, and at best the separate but equal facilities, whichhitherto 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 emergefrom this book.
　　It will be pointed out to me that many important facts and valuableresults about local fields can be proved in a fully algebraic context,without any use being made of local compacity, and can thus be shownto preserve their validity under far more general conditions. May I beallowed to suggest that I am not unaware of this circumstance, nor ofthe possibility of similarly extending the scope of even such global resultsas the theorem of Riemann-Roch? We are dealing here with mathematics,not with theology. Some mathematicians may think that they can gainfull insight into God's own way of viewing their favorite topic; to me,this has always seemed a fruitless and a frivolous approach. My intentionsin this book are more modest. I have tried to show that, from the pointof view which I have adopted, one could give a coherent treatment,logically and aesthetically satisfying, of the topics I was dealing with.I shall be amply rewarded if I am found to have been even moderatelysuccessful in this attempt.
　　Some of my readers may be surprised to find no explicit mention ofcohomology in my account of classfield theory. In this sense, while myapproach to number-theory may be called a "modern" one in the firsthalf of this book, it may well be described as thoroughly "unmodern" inthe second part. The sophisticated reader will of course perceive that acertain amount of cohomology, and in fact no more and no less than isrequired for the purposes of classfield theory, hides itself in the theoryof simple algebras. For anyone familiar with the language of "Galoiscohomology", it will be an easy and not unprofitable exercise to translateinto it some of the definitions and results of our Chapters IX, XII andXIII; in one or two places (the most conspicuous case being that of the"transfer theorem" in Chapter XII, 5), this even makes it possible tosubstitute more satisfactory proofs for ours. For me to develop such anapproach systematically would have meant loading a great deal ofunnecessary machinery on a ship which seemed well equipped for thisparticular voyage; instead of making it more seaworthy, it might havesunk it.
　　In charting my course, I have been careful to steer clear of the arith-metical theory of algebraic groups; this is a topic of deep interest, butobviously not yet ripe for book treatment. Partly for this reason, I haverefrained from discussing zeta-functions of simple algebras beyond whatwas needed for the sake of classfield theory. Artin's non-abelian L-func-tions have also been excluded; the reader of this book will find it easyto proceed to the study of Artin's beautiful papers on this subject andwill find himself well prepared to enjoy them, provided he has someknowledge of the representation theory of finite groups.
　　It remains for me to discharge the pleasant duty of expressing mythanks to David Cantor, who prepared from my lectures at PrincetonUniversity the set of notes which reappears here as Chapters I to VIIof this book (in many places with no change at all), and to Chevalley,who generously allowed me to make use of the above-mentioned manus-cript and expand it into Chapters Xll and XIlI. My thanks are alsodue to Iwasawa and Lazard. who read the book in manuscript and offeredmany suggestions for its improvement; to H. Pogorzelski, for his assis-tance in proofreading; to B. Eckmann, for the interest he took in itspublication; and to the staff of the Springer Verlag, and that of theZechnersche Buchdruckerei, for their expert cooperation and theirinvaluable help in the process of bringing out this volume.
　　Princeton, May 1967.
　　ANDRIE WEIL
　　Foreword to the second edition
　　The text of the first edition has been left unchanged. A few correc-tions, references, and some brief remarks, have been added as Notes atthe end of the book; the corresponding places in the text have beenmarked by a * in the margin. Somewhat more substantial additions willbe found in the Appendices, originally prepared for the Russian edition(M.I.R., Moscow 1971). The reader's attention should be drawn to thecollective volume: J.W.S. Cassels and A. FrShlich (edd.), AlgebraicNumber Theory, Acad. Press 1967, which covers roughly the sameground as the present book, but with far greater emphasis on the cohomo-logical aspects.
　　Princeton, December 1971.
　　ANDRE WEIL