From the reviews: "L.R. Shafarevich showed me the first edition [
] 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." Zentralblatt MATH
Recenzijos
"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
I. Elementary Theory.- 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.- 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.- 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.- IV. Adeles.-
1. Adeles of
A-fields.-
2. The main theorems.-
3. Ideles.-
4. Ideles of A-fields.-
V. Algebraic number-fields.-
1. Orders in algebras over Q.-
2. Lattices
over algebraic number-fields.-
3. Ideals.-
4. Fundamental sets.- VI. The
theorem of Riemann-Roch.- 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.- 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.- II.
Classfield Theory.- 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.- X.
Simple algebras over local fields.-
1. Orders and lattices.-
2. Traces
and norms.-
3. Computation of some integrals.- 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.- 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.- 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
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 III.
Shafarevitch's theorem.- Appendix IV. The Herbrand distribution.- Index of
definitions.
Biography of Andre Weil Andre Weil was born on May 6, 1906 in Paris. After studying mathematics at the Ecole Normale Superieure and receiving a doctoral degree from the University of Paris in 1928, he held professorial positions in India, France, the United States and Brazil before being appointed to the Institute for Advanced Study, Princeton in 1958, where he remained until he died on August 6, 1998. Andre Weil's work laid the foundation for abstract algebraic geometry and the modern theory of abelian varieties. A great deal of his work was directed towards establishing the links between number theory and algebraic geometry and devising modern methods in analytic number theory. Weil was one of the founders, around 1934, of the group that published, under the collective name of N. Bourbaki, the highly influential multi-volume treatise Elements de mathematique.
