本书作者Andre Weil为抽象代数几何及Abel簇的现代理论的研究奠定了基础，他的大多数研究工作都在致力于建立“数论”、“代数几何”之间的联系，以及发明解析数论的现代方法。Weil是1934年左右成立的Bourbaki学派的创始人之一，此学派以集体名称N.Bourbaki出版了有着很高影响力的多卷专著《数学的基础》。

本书为他的《基础数论》，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

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 class field 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

 1.The canonical pairing

 2.An elementary lemma

 3.Hasse's "law of reciprocity"

 4.Classfield theory for Q

 5.The Hilbert symbol

 6.The Brauer group of an A-field

 7.The Hilbert p-symbol

 8.The kernel of the canonical morpnism

 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