Basic Number Theory (Classics in Mathematics) (英語) ペーパーバック – 2013/10/4
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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
"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
As Weil says at the start of the book, it has few prerequisites in algebra or number theory, except that the reader is presumed familiar with the standard theorems on locally compact Abelian groups, and Pontryagin duality and Haar measures on those groups. This part is not a joke.
If you want to really understand class field theory this may be a good book. (I am reliably told it is.) But Weil deliberately avoids using many ideas that are now standard: geometric ideas such as group schemes, and especially cohomological methods.
Beginners studying algebraic numbers do not need this book. Weil recommends Hecke ALGEBRAIC NUMBERS for such readers, and that is a terrific book. To learn class field theory today you'd probably do better with and Cassels and Frohlich ALGEBRAIC NUMBER THEORY, which Weil also recommends in a note to the second edition of this book.