The study of number fields by embedding them in a local field is called local class field theory. Information about an equation in a local field may give information about the equation in a global field, such as the rational numbers or a number field (e.g., the Hasse principle).

Local class field theory is termed "local" because the local fields are localized at a prime ideal in the ring of algebraic integers. The methods of using class fields have developed over the years, from the Legendre symbol, to the group characters of Abelian extensions of a number field, and is applied to local fields.

 

Abelian Extension, Class Field, Field, Global Field, Hasse Principle, Local Field, Number Field, Unique Factorization