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

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{title2=$W_K$}
{wiki}

For a non-Archimedean local field $K$, the Weil group is the inverse image of the infinite cyclic subgroup generated by Frobenius under $\operatorname{Gal}(\overline K/K)\to\operatorname{Gal}(\overline k/k)\cong\widehat{\mathbb Z}$. It is dense in the absolute Galois group with its profinite topology.


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 Incoming links: Weil group