# The last part of Tate'sThesis

Is there a comprehensive reference dealing with the last part(theory in the large!) of Tate's thesis? Why is the group of S units modulo the roots of unity a free abelian group of rank m?

-
Something seems to have gone missing. –  joriki Apr 12 '11 at 19:05
Even clicking on edit does not show the missing part (which sometimes happens if you use < and >). –  Aryabhata Apr 12 '11 at 19:15
I believe the last part is mostly a summary of various –  user641 Apr 12 '11 at 19:44
Don't worry, you guys, I believe Jonathan will –  Guess who it is. Apr 13 '11 at 1:21
Sorry for the missing part... I had another question which thought I could work a lil more... I think the last part is not a summary but contains actual computations of zeta funcitons and relate them to the functional equation of Hecke. –  Jonathan Apr 13 '11 at 5:03

An adelic proof of the Unit Theorem for $S$-integer rings can be found as Theorem 8 in these notes. (You will find an attribution to Ramakrishnan-Valenza's Fourier Analysis on Number Fields. I have found this text to be readable and useful in general, but not completely reliable on the details. For instance, if memory serves I actually had to fix a false lemma in their text in order to carry out their proof, but having done that their proof is a very nice one.)