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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-123/4/2/c/solution