  Let F be a field of characteristic zero and let V be an irreducible
representation of SL_n(F). In this paper, we compute the first cohomology of
SL_n(F[t]) with coefficients in V. It agrees with H^1(SL_n(F),V) if V is not
the adjoint representation, while if V = Ad, the two groups differ by an
F-vector space X. We show that if n=2, X is infinite dimensional, while if n>2,
dim X = 1. We also study the abelianization of the kernel of the map
SL_n(F[t])-->SL_n(F) given by setting t=0, where now F is any field. We
conjecture that this abelianization is the adjoint representation sl_n(F) if
n>2 and F is finite, and prove this in the case n=3, F=F_2, F_3.
