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內容簡介

　　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 com- 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.

內頁插圖

目錄

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 0f 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

Chapter Ⅳ Adeles
1 Adeles of A-fields
2 The main theorems
3 Ideles
4 Ideles of A-fields

Chapter Ⅴ Algebraic number-fields
1， Orders in algebras over Q
2 Lattices over algebraic number-fields
3 Ideals
4 Fundamental sets

Chapter Ⅵ The theorem of Riemann-Roch
Chapter Ⅶ Zeta-functions of A-fields
1 Convergence of Euler products
2 Fourier transforms and standard functions
3 Quasicharacters
4 Quasicharacters of A-fields
5 The functional equation
6 The Dedekind zeta-function
7 L-functions
8 The coefficients of the L-series

Chapter Ⅷ Traces and norms
1 Traces and norms in local fields
2 Calculation of the different
3 Ramification theory
4 Traces and norms in A-fields
5 Splitting places in separable extensions
6 An application to inseparable extensions

PART Ⅱ CLASSFIELD THEORY
Chapter IX Simple algebras
1 Structure of simple algebras
2 The representations of a simple algebra
3 Factor-sets and the Brauer group
4 Cyclic factor-sets
5 Special cyclic factor-sets

Chapter Ⅹ Simple algebras over local fields
1 Orders and lattices
2 Traces and norms
3 Computation of some integrals

Chapter Ⅺ Simple algebras over A-fields
1. Ramification
2. The zeta-function of a simple algebra
3. Norms in simple algebras
4. Simple algebras over algebraic number-fields . .

Chapter Ⅻ. Local classfield theory
1. The formalism of classfield theory
2. The Brauer group of a local field
3. The canonical morphism
4. Ramification of abelian extensions
5. The transfer

Chapter XIII. Global classfield theory
I. The canonical pairing
2. An elementary lemma
3. Hasses "law of reciprocity" .
4. Classfield theory for Q
5. The Hiibert symbol
6. The Brauer group of an A-field
7. The Hilbert p-symbol
8. The kernel of the canonical morphism
9. The main theorems
10. Local behavior of abelian extensions
11. "Classical" classfield theory
12. "Coronidis loco".
Notes to the text
Appendix Ⅰ. The transfer theorem
Appendix Ⅱ. W-groups for local fields
Appendix Ⅲ. Shafarevitchs theorem
Appendix Ⅳ. The Herbrand distribution
Index of definitions

前言/序言

　　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. 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 magicians 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， for bringing them all together under one roof and making them cooperate for 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 this road. Once the presence of the real field， albeit at infinite distance， ceases to be regarded as a necessary ingredient in the arithmeticians brew.

基礎數論（英文版） [Basic Number Theory] 下載 mobi epub pdf txt 電子書

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藉助微積分及復分析（即復變函數）來研究關於整數的問題，主要又可以分為乘性數論與加性數論兩類。乘性數論藉由研究積性生成函數的性質來探討質數分布的問題，其中質數定理與狄利剋雷定理為這個領域中最著名的古典成果。加性數論則是研究整數的加法分解之可能性與錶示的問題，華林問題是該領域最著名的課題。

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