 Weil-group topology

The topology of the absolute Weil group $W_K$ of a non-Archimedean local field makes the inertia group open with its profinite topology, and makes the quotient $W_K/I_K\cong \mathbb{Z}$ discrete. In particular inertia cosets are open. This topology is finer than the subspace topology inherited from the absolute Galois group: the induced topology on the integer quotient would otherwise be its nondiscrete profinite topology.

For a finite Galois extension $L/K$, the Relative Weil group instead has the open subgroup $L^{\times }$ with its usual topology and finite quotient $\mathrm{Gal}(L/K)$. Keeping the absolute and relative constructions separate prevents a finite relative Galois quotient from being confused with the infinite residue-action quotient.^[ref]