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 Q_p the p-adic numbers, or power series in a complex variable.

function field | global field | Hasse principle | local class field theory | number field | p-adic number | valuation