Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 123 / 4 / 2 / c / Solution

The idele norm has $v$-componentFor the principal idele associated with $\alpha \in L^{\times }$, compatibility of local and global field norms givesThe right side is the principal idele of the element $N_{L/K}(\alpha )\in K^{\times }$.