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

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- Solution a

Solution

The idele group is the restricted product

I_{K} = \prod_{v}^{'} K_{v}^{\times}

over all places, with respect to

O_{v}^{\times}

at finite places. Thus an idele has a nonzero component in every completion and belongs to

O_{v}^{\times}

at all but finitely many finite places.

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