Higher Local Class Field Theory
For a higher dimensional local field there is a higher local reciprocity map which describes abelian extensions of the field in terms of open subgroups of finite index in the Milnor K-group of the field. Namely, if is an -dimensional local field then one uses or its separated quotient endowed with a suitable topology. When the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if . Higher dimensional class field theory was pioneered by A.N. Parshin in positive characteristic and K. Kato, I. Fesenko, Sh. Saito in the general case.
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