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I was wondering if for a global field (function or number field) $K$, is $C_K$ Hausdorff?

Thank you

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Yes it is Hausdorff, because the idele group $I_K$ of a global number field $K$ is a locally compact, topological group and the subgroup $K^\times$ of principial idèles is discrete and hence closed. So denoting by $C_K$ the idèle class group of $K$, it is a locally compact, hausdorff topological group. You can find proofs for all this in chapter VI, §1 of Neukirch's book "Algebraic Number Theory".

Enjoy your weekend, Tom