In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Qp, or the field of formal Laurent series Fq over a finite field Fq. Wikipedia