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Most Helpful Customer Reviews
53 of 55 people found the following review helpful:
5.0 out of 5 stars
A warning to beginners,
By Colin McLarty (Chardon, OH USA) - See all my reviews
This review is from: Basic Number Theory (Classics in Mathematics) (Paperback)
Experts find this a very good book, and I rate it on their advice. But others need to understand that Weil is making a bit of a joke with the title. This book is "basic" in the sense that it proves the theorems that Weil feels organize and clarify all of number theory--the "basic" theorems in that sense. It is an introduction to class field theory. 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.
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