Weil group ($W_K$)

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

For a finite Galois extension $L/K$ of non-Archimedean local fields, the Relative Weil group isHere $W_L$ is the subgroup of the absolute Weil group corresponding to $L$. Local Artin reciprocity identifies its abelianization with $L^{\times }$, giving an extensionIts extension class in $H^2(\mathrm{Gal}(L/K),L^{\times })$ is the local fundamental class. The subgroup $L^{\times }$ is open and has its usual local-field topology. For $L=K$, this recovers $W_K^{\mathrm{ab}}\cong K^{\times }$. This relative extension has a different kernel from the residue-action description of the absolute Weil group.^[ref]

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]