Norm group
In number theory, a norm group is a group of the form  where  is a finite abelian extension of nonarchimedean local fields. One of the main theorems in local class field theory states that the norm groups in  are precisely the open subgroups of  of finite index. See also 
Takagi existence theorem

References 
J.S. Milne, Class field theory. Version 4.01. Algebraic number theory