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

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- Solution c

Solution

The idele norm has

v

-component

(N_{L / K} x)_{v} = \prod_{w ∣ v} N_{L_{w} / K_{v}} (x_{w}) .

For the principal idele associated with

α \in L^{\times}

, compatibility of local and global field norms gives

N_{L / K} ((α)_{w}) = (N_{L / K} (α))_{v} .

The right side is the principal idele of the element

N_{L / K} (α) \in K^{\times}

.

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