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I've been reading about adeles and ideles and many authors like Milne and Lang spend some time discussing compactness results related to them. This seemed to me more like a technical point until I found this question in the algebraic number theory collection of old questions for Princeton's generals:

What results do the major compactness theorems about adeles and ideles imply?

Are these theorems directly implying some number theoretical results through class field theory?

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Note that some sources (Neukirch, at least) prove compactness of the idele class group as a consequence of the finiteness of the class group and Dirichlet's Units Theorem, but it can be proven directly. –  B R Nov 10 '11 at 9:05

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If I recall correctly, in Cassels and Frohlich, you can find proofs showing that certain compactness results for ideles imply these classical results for number fields: (1) the ideal class group is finite; (2) Dirichlet's Unit Theorem.

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I recall Tom Weston's notes were particularly readable regarding the Unit Theorem (the finiteness of the ideal class group is pretty obvious). See Section 10. Link: math.umass.edu/~weston/oldpapers/idele.pdf –  B R Nov 10 '11 at 9:00
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I think this is it. A google search for those results also come up with a REU paper from Chicago which basically proves just that using adeles and ideles. It can be found here: math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Petkov.pdf –  pki Nov 10 '11 at 17:33
    
Combining the comment of @BR, and the answer of Ted, is it true that one can state the equivalence of the compactness and the two conditions listed there? It seems natural in the writings of both authors, but not for me, sorry, as I am not well versed at it now. –  awllower Nov 13 '11 at 14:33
    
awllower, yes, the finiteness of the ideal class group and the units theorem (proven with classical methods) prove the compactness of the norm-one idele class group, and the compactness of the norm-one idele class group, proven using measure theoretic methods proves finiteness of the ideal class group and the units theorem. On the other hand, the compactness result does generalize to division algebras, where there is not an obvious version of the ideal class group, etc (See Weil's Basic Number Theory). –  B R Nov 13 '11 at 17:55

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