Local Field

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  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 numbers, or power series in a complex variable.

REFERENCES:

Iyanaga, S. and Kawada, Y. (Eds.). "Local Fields." §257 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 811-815, 1980.