Relative Weil group
= Relative Weil group
{c}
{title2=$W(L/K)$}
For a finite <Galois extension> $L/K$ of non-Archimedean <local fields>, the <Relative Weil group> is
$$
W_{L/K}=W_K/\overline{[W_L,W_L]}.
$$
Here $W_L$ is the subgroup of the absolute <Weil group> corresponding to $L$. <Local Artin reciprocity> identifies its abelianization with $L^\times$, giving an extension
$$
1\longrightarrow L^\times\longrightarrow W_{L/K}\longrightarrow\operatorname{Gal}(L/K)\longrightarrow1.
$$
Its extension class in $H^2(\operatorname{Gal}(L/K),L^\times)$ 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.<Arithmetic Duality Theorems by James S. Milne (2006)>[ChapterI, §8, printed p.102, Weil groups]{ref}