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

The idele group is the restricted productover 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.