The purpose of this paper is two fold. Firstly, it gives a thorough introduction to the topological cyclic homology theory, which to a ring R associates a spectrum TC(R). We determine TC(k) and TC(k[x]/(x^2)) where k is a perfect field of positive characteristic and k[x]/(x^2) its dual numbers, and sets the stage for further calculations. Secondly, we show that the cyclotomic trace from Quillen's K(R) to TC(R) becomes a homotopy equivalence after p-adic completion when R is a finite algebra over the Witt vectors W(k) of a perfect field of characteristic p>0. This involves a recent relative result of R. McCarthy, the calculation of TC(k) and Quillen's theorem about K(k), and continuity results for TC(R) and K(R), the latter basically due to Suslin and coworkers. In particular, we obtain a calculation of the tangent space of K(k), i.e. the homotopy fiber of the map from K(k[x]/(x^2)) to K(k) induced from the map which maps x to zero.