Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 136 / 5 / a / Solution

Restriction to the maximal unramified extension gives a surjectionThe abelian Weil group is the inverse image of the dense subgroup $\mathbb{Z}\subset \hat{\mathbb{Z}}$ generated by Frobenius:It contains the full inertia kernel. Since $\mathbb{Z}$ is dense in $\hat{\mathbb{Z}}$, the inverse image is dense in $\mathrm{Gal}({\mathbb{Q}}_p^{\mathrm{ab}}/{\mathbb{Q}}_p)$ with its profinite topology.