This paper gives a modern proof of Quillen's calculation of the
K-theory of finite fields, which makes full use of the Gabber rigidity
theorem and the homotopy theory of simplicial presheaves. The method
is to give a direct proof of the assertion that the K-theory of a finite
field coincides with the stable homotopy groups of the homotopy fixed
points for the action of the Frobenius on the K-theory spectrum of the
algebraic closure, in non-negative degrees.
This paper has appeared in print, and the corresponding dvi file has
been removed from this archive. Here is the reference:
J.F. Jardine: "The K-theory of Finite Fields, Revisited", K-Theory 7
(1993), 579-595.