  We determine the structure modulo p of the de Rham-Witt complex of a smooth
scheme X over a discrete valuation ring of mixed characteristic with log-poles
along the special fiber Y and show that the sub-sheaf fixed by the Frobenius is
isomorphic to the sheaf of p-adic vanishing cycles. We use this together with
results of the second author and Madsen to evaluate the $K$-theory with finite
coefficients of the quotient field K of the henselian local ring of X at a
generic point of Y. The result affirms the Lichtenbaum-Quillen conjecture for
the field K.
