Norm subgroup of a local field extension ($N_{L/K}(L^{\times })$)

For a finite extension of local fields $L/K$, its norm subgroup is the image of the field norm $N_{L/K}:L^{\times }\to K^{\times }$. For an abelian extension, it is the kernel of the induced local Artin map to $\mathrm{Gal}(L/K)$.