Norm group

In number theory, a norm group is a group of the form ${\displaystyle N_{L/K}(L^{\times })}$ where ${\displaystyle L/K}$ is a finite abelian extension of nonarchimedean local fields, and ${\displaystyle N_{L/K}}$ is the field norm. One of the main theorems in local class field theory states that the norm groups in ${\displaystyle K^{\times }}$ are precisely the open subgroups of ${\displaystyle K^{\times }}$ of finite index.