In number theory, a norm group is a group of the form N L / K ( L × ) {\displaystyle N_{L/K}(L^{\times })} where L / K {\displaystyle L/K} is a finite abelian extension of nonarchimedean local fields, and N L / K {\displaystyle N_{L/K}} is the field norm. One of the main theorems in local class field theory states that the norm groups in K × {\displaystyle K^{\times }} are precisely the open subgroups of K × {\displaystyle K^{\times }} of finite index.
See also Takagi existence theorem
References J.S. Milne, Class field theory. Version 4.01.