The goal

There is an idelic compactness statement which encapsulates both finiteness of the class group of a number field and Dirichlet's units theorem about the rank of the unit group. In fact there is even a noncommutative version of this statement. In John Voight's book (Voight 2021) this is Main Theorem 27.6.14(a) and Voight calls it Fujisaki’s lemma. I know nothing of the history but I'm happy to adopt this name. In the quaternion algebra miniproject we will use this compactness result to prove finite-dimensionality of a space of quaternionic modular forms.

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