Restriction to the maximal unramified extension gives a surjection
$$
\operatorname{Gal}(\mathbb Q_p^{\mathrm{ab}}/\mathbb Q_p)
\longrightarrow
\operatorname{Gal}(\overline{\mathbb F}_p/\mathbb F_p)
\cong\widehat{\mathbb Z}.
$$
The abelian <Weil group> is the inverse image of the dense subgroup $\mathbb Z\subset\widehat{\mathbb Z}$ generated by Frobenius:
$$
W(\mathbb Q_p^{\mathrm{ab}}/\mathbb Q_p)
=\{g:g|_{\overline{\mathbb F}_p}=\operatorname{Frob}_p^n\text{ for some }n\in\mathbb Z\}.
$$
It contains the full inertia kernel. Since $\mathbb Z$ is dense in $\widehat{\mathbb Z}$, the inverse image is dense in $\operatorname{Gal}(\mathbb Q_p^{\mathrm{ab}}/\mathbb Q_p)$ with its profinite topology.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-136/5/a/solution