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

Consider a complete discrete valuation ring O with quotient field F and finite residue field. Then the inclusion map O↪F induces a map Kˆ⁎MO→Kˆ⁎MF on improved Milnor K-theory. We show that this map is an isomorphism in degrees bigger or equal to 3. This implies the Gersten conjecture for improved Milnor K-theory for O. This result is new in the p-adic case.