  A proof of the vanishing of the third cohomology group of the Witt algebra
with values in the adjoint module is given. Moreover, we provide a sketch of
the proof of the one-dimensionality of the third cohomology group of the
Virasoro algebra with values in the adjoint module. The proofs given in the
present article are completely algebraic and independent of any underlying
topology. They are a generalization of the ones provided by Schlichenmaier, who
proved the vanishing of the second cohomology group of the Witt and the
Virasoro algebra by using purely algebraic methods. In the case of the third
cohomology group though, extra difficulties arise and the involved proofs are
distinctly more complicated. The first cohomology group can easily be computed;
we will give an explicit proof of its vanishing in the appendix, in order to
illustrate our techniques.
