Norm group of an abelian extension
= Norm group of an abelian extension
{title2=$N_{L/K}C_L$}
For a finite abelian extension $L/K$, the norm group is the image of the idele-class norm $C_L\to C_K$. Class field theory gives $C_K/N_{L/K}C_L\cong\operatorname{Gal}(L/K)$.