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Consider a complete discrete valuation ring $\mathcal{O}$ with quotient field$F$ and finite residue field. Then the inclusion map $\mathcal{O}\hookrightarrow F$ induces a map $\hat{\mathrm{K}}^\mathrm{M}_*\mathcal{O} \to\hat{\mathrm{K}}^\mathrm{M}_*F$ on improved Milnor K-theory. We show that thismap is an isomorphism in degrees bigger or equal to 3. This implies the Gerstenconjecture for improved Milnor K-theory. This result is new in the $p$-adiccase.

Milnor K-theory of complete discrete valuation rings with finite residue fields