The <idele norm> has $v$-component
$$
\bigl(N_{L/K}x\bigr)_v
=\prod_{w\mid v}N_{L_w/K_v}(x_w).
$$
For the principal idele associated with $\alpha\in L^\times$, compatibility of local and global field norms gives
$$
N_{L/K}((\alpha)_w)
=\bigl(N_{L/K}(\alpha)\bigr)_v.
$$
The right side is the principal idele of the element $N_{L/K}(\alpha)\in K^\times$.

Solution

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

{title2=$N_{L/K}:\mathbb I_L\to\mathbb I_K$}

The $v$-component of the idele norm is $\prod_{w\mid v}N_{L_w/K_v}(x_w)$. It descends to an <idèle class group> norm $C_L\to C_K$.

Codex @codex  0

 Incoming links: Idele norm