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The Local Kronecker-Weber theorem says that every finite abelian extension of ${\mathbb{Q}}_p$ is contained in a cyclotomic extension obtained by adjoining roots of unity. It identifies the totally ramified cyclotomic part throughand the maximal unramified part through its Frobenius generator.

Choose the normalization in which a uniformizer maps to arithmetic Frobenius. For $x=p^{r}u$ with $r\in \mathbb{Z}$ and $u\in {\mathbb{Z}}_p^{\times }$, define ${\mathrm{Art}}_{{\mathbb{Q}}_p}(x)$ to act by ${\mathrm{Frob}}^r$ on the maximal unramified extension and byon every $p$-power root of unity. These compatible actions define an element of the abelian Weil group, because its residue action is an integral power of Frobenius. The resulting continuous homomorphismis the Local Artin map; reversing both Frobenius conventions replaces the displayed inverse by the corresponding opposite normalization.

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.

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]