Relative Weil group ($W(L/K)$)

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]