We prove that for W_2(F_q) the Witt vectors of length two over the finite field F_q, we have

K_3(W_2(F_{p^f}))=(Z/p^2)^f + Z/(p^{2f}-1)in characteristic at least 5 and

K_3(W_2(\F_{3^f}))=(Z/9)^{f-1} + (Z/3)^2 + Z/(3^{2f}-1)for f prime to 3. The result is proved by using the equality

K_3(W_2(\F_q))=H_3(SL(W_2(\F_q))and calculating the right term with a group homology spectral sequence. Some terms in the spectral sequence are determined by using the action of the outer automorphism of SL on the homology groups and recent results on K-groups of local rings and the ring of dual numbers over finite fields. In characteristic 3 we have to calculate an explicit differential in the spectral sequence. This is done with some Mathematica programs, which will be made available if requested via email.

A revised version of this preprint was placed here November 27, 1997.

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Thomas Geisser <geisser@math.harvard.edu>