  We give upper bounds on the numbers of various classes of polynomials
reducible over the integers and over integers modulo a prime and on the number
of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic
polynomials, and on polynomials with non-generic Galois groups. We use our
result to show that a random (in the appropriate sense) element of the mapping
class group of a closed surface is pseudo-Anosov, and that a random
automorphism of a free group is strongly irreducible (aka irreducible with
irreducible powers).
  We also give a necessary condition for all powers of an algebraic integers to
be of the same degree, and give a simple proof (in the Appendix) that the
distribution of cycle structures modulo a prime p for polynomials with a
restricted coefficient is the same as that for general polynomials.
