  We discuss several enumerative results for irreducible polynomials of a given
degree and pairs of relatively prime polynomials of given degrees in several
variables over finite fields. Two notions of degree, the {\em total degree} and
the {\em vector degree}, are considered. We show that the number of
irreducibles can be computed recursively by degree and that the number of
relatively prime pairs can be expressed in terms of the number of irreducibles.
We also obtain asymptotic formulas for the number of irreducibles and the
number of relatively prime pairs. The asymptotic formulas for the number of
irreducibles generalize and improve several previous results by Carlitz, Cohen
and Bodin.
