The <idele group> is the restricted product
$$
\mathbb I_K=\prod_v'K_v^\times
$$
over all places, with respect to $\mathcal O_v^\times$ at finite places. Thus an idele has a nonzero component in every completion and belongs to $\mathcal O_v^\times$ at all but finitely many finite places.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 4 / a / Solution

{title2=$\mathbb I_K$}
{wiki=Idèle_class_group}

The idele group is the restricted product $\prod_v'K_v^\times$ with respect to $\mathcal O_v^\times$ at finite places.


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 Incoming links: Idele group