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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.
See also Hasse Principle, Valuation
References
Iyanaga, S. and Kawada, Y. (Eds.). ``Local Fields.'' §257 in
Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 811-815, 1980.