# Finiteness of class group in idelic language

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!

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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.