#### DOCUMENTA MATHEMATICA, Vol. 5 (2000), 151-200

Jinya Nakamura

On the Milnor $K$-Groups of Complete Discrete Valuation Fields

For a discrete valuation field $K$, the unit group $K^{\times}$ of $K$ has a natural decreasing filtration with respect to the valuation, and the graded quotients of this filtration are given in terms of the residue field. The Milnor $K$-group $\MK_q(K)$ is a generalization of the unit group, and it also has a natural decreasing filtration. However, if $K$ is of mixed characteristics and has an absolute ramification index greater than one, the graded quotients of this filtration are not yet known except in some special cases. \par

The aim of this paper is to determine them when $K$ is absolutely tamely ramified discrete valuation field of mixed characteristics $(0,p >2)$ with possibly imperfect residue field. \par

Furthermore, we determine the kernel of the Kurihara's $\MK_q$-exponential homomorphism from the differential module to the Milnor $K$-group for such a field.

1991 Mathematics Subject Classification: 19D45, 11S70

Keywords and Phrases: The Milnor $K$-group, Complete Discrete Valuation Field, Higher Local Class Field Theory

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