# Norm group

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In [number theory](/source/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](/source/Abelian_extension) of nonarchimedean [local fields](/source/Local_field), and N L / K {\displaystyle N_{L/K}} is the [field norm](/source/Field_norm). One of the main theorems in [local class field theory](/source/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](/source/Index_(group_theory)).

## See also

- [Takagi existence theorem](/source/Takagi_existence_theorem)

## References

- J.S. Milne, *Class field theory.* Version 4.01.

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