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I'm searching for a place where is developed all the machinery of adeles and ideles of a number field; the functional equation for the zeta functions is gained by idelic integration, and it's seen the connection between the compactness(and the finitness of measure, that comes from the functional equation) of the quotient (ideles of absolute value 1)/(nonzero rationals) and the two finitness theorem of class group and Dirichlet unit theorem.

I would like somewhere, where is proved that:

1)The only valutations for number field archimedean are the $[K:\mathbb{Q}]$ given by the immersion composed with standard absolute value(the non archimedean are easy to work out as the usual p-adic relatevily to the primes of the ring of integers).

2)The standard fourier analysis is developed for adeles of number field(at least i've seen in the rational case,the deduction of the functional equation for the zeta, using self duality of this group, discrete immersion of Q, and fourier expansion by characters e(ax) for Q periodic function, a in Q, and so Poisson summation formula, that is(as it seems to me)the major tool that pushes the computation to the functional equation).

3)The basic theorems of algebraic number theory are proved by the finitness of (ideles of absolute value 1)/(non zero rational)

I think i've worked the second thing by analogy with the rational case that i've already seen, and 3) just by inspection because once someone told you that these things are related is just clear to see how.

But i would really like a reference for all this stuff: also more of them are welcome, since i'm curious to see people's intuition about this stuff, since i'm studying it on my own. Thanks

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  • $\begingroup$ Neukirch's book is excellent. Manin-Panchishkin is also great, although it's much longer and covers a lot of elementary stuff. $\endgroup$ – Justin Campbell Feb 8 '14 at 3:50
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    $\begingroup$ For 1) this is just the standard Ostrowski theorem for number fields, you can find it anywhere. For the rest, have you tried looking at Tate's thesis? $\endgroup$ – Alex Youcis Feb 8 '14 at 3:51
  • $\begingroup$ @JustinCampbell:many thanks for the suggestions, i'll have a look. $\endgroup$ – Donald Feb 8 '14 at 3:53
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    $\begingroup$ @AlexYoucis: thanks for the suggestion.I haven't read Tate thesis, but it seems that would be a good idea to do so.p.s:i had seen Ostrowski theorem worked out just for the rational case, i didn't know is named to him also this more general case, thanks for the info. $\endgroup$ – Donald Feb 8 '14 at 3:58
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    $\begingroup$ A book that develops algebraic number theory from analysis on topological groups is Weil's "Basic Number Theory". You might also like Larry Goldstein's "Analytic Number Theory". $\endgroup$ – KCd Feb 8 '14 at 4:04
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What you want is the wonderful book Fourier analysis on number fields by Ramakrishnan and Valenza.

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