Basic number theory.
Reprint of the 2nd ed. 1973.

*(English)*Zbl 0823.11001
Classics in Mathematics. Berlin: Springer-Verlag. xviii, 312 p. (1995).

The first edition of Weil’s book was reviewed in (1967; Zbl 0176.336) and the second in (1973; Zbl 0267.12001). Therefore I restrict here to some general remarks.

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 “basic 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.

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 “basic 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.

Reviewer: H.Koch (Berlin)

##### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11Rxx | Algebraic number theory: global fields |

11R37 | Class field theory |

11R58 | Arithmetic theory of algebraic function fields |