Polynomials, Permutations, Prime Ideals, and factoring Polynomials modulo p

The talk began as an attempt to answer the following question - if f(x) is an irreducible polynomial with integer coefficients, does it remain irreducible modulo p for infinitely many primes p? It turns out that the answer is sometimes yes and sometimes no. One can give the answer satisfactorily and even give the Dirichlet density of the set of p for which the reduction is irreducible.