On the Milnor $K$-groups of complete discrete valuation fields

Abstract

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 $K_q^M(K)$ 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.