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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.2011-11-10

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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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    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).2011-11-13