A field which is complete with respect to a discrete valuation is called a local field if its field of residue classes is finite. The Hasse principle is one of the chief applications of local field theory. A local field with field characteristic p > 0 is isomorphic to the field of power series in one variable whose coefficients are in a finite field. A local field of characteristic zero is either the p-adic Number, or power series in a complex variable.

 

Function Field, Global Field, Hasse Principle, Local Class Field Theory, Number Field, p-adic Number, Valuation