中文名: 基礎數論
原名: Basic Number Theory
作者: Andre Weil
資源格式: PDF
版本: 掃描版
出版社: Springer
書號: 3540586555
發行時間: 1995年
地區: 美國
語言: 英文
簡介:


內容簡介:
本書是一部學習“類域論”的非常好的教材。學習本書不需要任何數論的基礎知識，但需要熟知局部緊Abel環，Pontryagin對偶性以及群上的Haar測度的標准定理。此外，本書不適於代數數論的初學者使用。
Review
"L.R. Shafarevich showed me the first edition in autumn 1967 in Moscow and said that this book will be from now on the book about class field theory. In fact it is by far the most complete treatment of the main theorems of algebraic number theory, including function fields over finite constant fields, that appeared in book form. The theory is presented in a uniform way starting with topological fields and Haar measure on related groups, and it includes not only class field theory but also the theory of simple algebras over local and global fields, which serves as a foundation for class field theory. The spirit of the book is the idea that all this is asic number theory' about which elevates the edifice of the theory of automorphic forms and representations and other theories.
To develop this basic number theory on 312 pages efforts a maximum of concentration on the main features. So, there is absolutely no example which illustrates the rather abstract material and brings it nearer to the heart of the reader.
This is not a book for beginners. This book is written in the spirit of the early forties and just this makes it a valuable source of information for everyone who is working about problems related to number and function fields."
Zentralblatt MATH, 823
內容截圖:

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 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