  We study the problem of consistent interactions for spin-3 gauge fields in
flat spacetime of arbitrary dimension n>3. Under the sole assumptions of
Poincar\'e and parity invariance, local and perturbative deformation of the
free theory, we determine all nontrivial consistent deformations of the abelian
gauge algebra and classify the corresponding deformations of the quadratic
action, at first order in the deformation parameter. We prove that all such
vertices are cubic, contain a total of either three or five derivatives and are
uniquely characterized by a rank-three constant tensor (an internal algebra
structure constant). The covariant cubic vertex containing three derivatives is
the vertex discovered by Berends, Burgers and van Dam, which however leads to
inconsistencies at second order in the deformation parameter. In dimensions n>4
and for a completely antisymmetric structure constant tensor, another covariant
cubic vertex exists, which contains five derivatives and passes the consistency
test where the previous vertex failed.
