TITLE: Finiteness of class group in idelic language
QUESTION [5 upvotes]: How should I understand the compactness of $A_{\mathbb{K}}^1/\mathbb{K}^{\times}$ in classical non-idelic language? I suppose the notations are standard, but just for completeness,

$K$ is a global field; 
$A_{\mathbb{K}}$ is the adele ring;
$A_{\mathbb{K}}^1$ is the kernel of the content map, defined on the idele
ring by multiplying the normalized absolute values at each place.
$K^{\times}$ embeds into $A_{\mathbb{K}}^1$ diagonally.

Motivation:
I am trying to compare the idelic proof in Cassels-Frohlich to the classical proof involving Minkowski bound. I have been able to translate part of the story, e.g. the lemma on p.66 of Cassels-Frohlich seems to be an analogue of Minkowski's convex body theorem. However, the crux of the idelic proof seems to be the compactness of the group mentioned above, and I have been unable to see what it corresponds to in the classical case. 
Thanks!

REPLY [4 votes]: The compactness of the norm one idele class group for a global field $K$ (Fujisaki's Lemma) is in fact equivalent to the finiteness of the ideal class group (of any one $S$-integer ring) and the Dirichlet unit theorem (of any one $S$-integer ring), including the precise computation of the free rank of the unit group.  
This perspective is given a very nice treatment in the number field case in this brief note of Paul Garrett.