Local class field theory describes finite abelian extensions of local fields using their multiplicative groups. 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).
Nonarchimedean local fields arise as completions of global fields at nonarchimedean absolute
values. Examples include finite extensions of 
and the formal Laurent
series fields 
. 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 are applied
to local fields.
