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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.

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    $\begingroup$ 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)? $\endgroup$ Jan 30 '11 at 16:54
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    $\begingroup$ 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. $\endgroup$ Jan 30 '11 at 18:23
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    $\begingroup$ 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 $\endgroup$
    – Emerton
    Jan 31 '11 at 0:34
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    $\begingroup$ 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. $\endgroup$
    – B R
    Feb 1 '11 at 7:43
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    $\begingroup$ Joël, it must be Dirichlet's Vorlesungen (with many Supplements by Dedekind), partially translated into English by John Stillwell. $\endgroup$ May 10 '13 at 2:36
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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
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  • $\begingroup$ Indeed this book is pretty great! $\endgroup$
    – awllower
    Feb 3 '11 at 1:19
  • $\begingroup$ 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. $\endgroup$ Feb 3 '11 at 14:50
  • $\begingroup$ 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. $\endgroup$
    – awllower
    Feb 3 '11 at 17:49
  • $\begingroup$ @Snark, thanks for your answer, and may I ask that is the book by Cassels and Froehlich in fact by Taylor and Froelich which is the only one I by far can get, thanks. $\endgroup$
    – awllower
    Feb 21 '11 at 14:13
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    $\begingroup$ 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! $\endgroup$ Feb 21 '11 at 20:26

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