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Is there any book explaining in detail the book "Basic Number Theory" by Andre Weil as Dirichlet did to "Disquisitiones Arithmeticae"? This is because I have read the two books mentioned above and I hope there will be one.

edited May 9 at 7:31

GH

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awllower 67%

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Gauss' book is, by those who tried to read it, reported as very hard to read, in contrast to Weil's book. Perhaps that comes from Gauss' book being a kind of starting point of a development, whereas Weil's tells a story after it is ended (and calls it "Basic" to distiguísh it from the new stories)? – Thomas Riepe Jan 30 2011 at 16:54

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Gauss's book was difficult to understand in the early 1800's. A lot of the difficult stuff is perfectly explained by Flath's Introduction to Number Theory. – Franz Lemmermeyer Jan 30 2011 at 18:23

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Dear Thomas, My own experience is that Tate's "number theory background" is best approach from a position of some sophistication, much more than say is required for Weil's "Basic Number Theory" or Tate's thesis. Best wishes, Matthew – Emerton Jan 31 2011 at 0:34

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Since no one else has mentioned it, Ramakrishnan and Valenza's book "Fourier Analysis on Number fields" covers much of the critical material in the first part of Weil's book, as well as the harmonic analysis it assumes (Haar measure and Pontryagin duality). To a certain extent, Basic Number Theory is a proof-of-concept: in the first part, Weil does algebraic number theory without algebra (using measure theory and topology), and in the second part, he does class field theory only using simple algebras. – BR Feb 1 2011 at 7:43

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Joël, it must be Dirichlet's Vorlesungen (with many Supplements by Dedekind), partially translated into English by John Stillwell. – Chandan Singh Dalawat May 10 at 2:36

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Julien Puydt · Accepted Answer · 2011-02-04 08:07:25Z UTC

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Indeed Ramakrishnan and Valenza 's book is a pretty good reference.

Perhaps we could give more specific answers if you were more precise about exactly where your difficulties are?

EDIT: Since we've been given precisions in the comments below, I can confirm R&V's book will nicely do for the basics of the theory ; to get further, from the top of my head, you'll want to have a go with :

Cassels and Froehlich (editors), "Algebraic number theory"
Serre, "Corps locaux" (translated)
Neukirch, "Class field theory"
J.S.Milne's notes on class field theory on his website

edited Feb 4 2011 at 8:07

Julien Puydt

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Indeed this book is pretty great! – awllower Feb 3 2011 at 1:19

Yes, but it was already mentioned by BR just above ; it would be more interesting to know if the question was more about what adeles are, about the zeta functions of number fields or class field theory. – Julien Puydt Feb 3 2011 at 14:50

Basically, I would like to find books on the notions of those topics mentioned by A.W. as clear as possible in the book since as far as I am concerned he preassumes that the readers already know the topics enough such that sometimes I feel like that I cannot grab the essence of notions , and besides, the exercises are absence in the book. This implies that I desire a book about CFT which requires as less prerequisites as possible. Thanks in any way. – awllower Feb 3 2011 at 17:49

Thanks in the end. – awllower Feb 16 2011 at 1:11

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Cassels and Froehlich are the editors ; there are various authors (Atiyah, Birch, Cassels, Froehlich, Gruenberg, Hasse, Heilbronn, Hoechsmann, Kneser, Roquette, Serre, Swinnerton-Dyer, Tate and Wall) ; those are the proceedings of a conference in Brighton in 1965. If you have problems finding the book, J.S. Milne's notes are easy to get -- you obviously have internet access! – Julien Puydt Feb 21 2011 at 20:26

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Is there any book explaining in detail the book “Basic Number Theory” by Andre Weil as Dirichlet did to “Disquisitiones Arithmetica” by Gauss?

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