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 Coevally, of prewar 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 complete account of the main features of 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 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.

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. Hasse's "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. Shafarevitch's theorem

Appendix Ⅳ. The Herbrand distribution

Index of definitions

Andre Weil 1906年5月6日齣生於巴黎，1928年於巴黎大學獲得博士學位，他曾先後在印度，法國，美國及巴西等國執教，1958年來到普林斯頓高等研究院從事研究工作，離休後現任該處終身教授。

Andre Weil的工作為抽象代數幾何及Abel簇的現代理論的研究奠定瞭基礎，他的大多數研究工作都在緻力於建立“數論”、“代數幾何”之間的聯係，以及發明解析數論的現代方法。Weil是1934年左右成立的Ｂourbaki學派的創始人之一，此學派以集體名稱N.Bourbaki齣版瞭有著很高影響力的多捲專著《數學的基礎》。