IRREDUCIBILITY OF ITERATES OF POST-CRITICALLY FINITE QUADRATIC POLYNOMIALS OVER Q

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials f(x) is an element of Z[x] with an iterate reducible module every prime, but all of whose iterates are irreducible over Q. In particular, we obtain infinitely many new examples of the phenomenon studied in [5]. While doing this, we also find, up to three possible exceptions, all integers a such that all iterates of the quadratic polynomial (x + a)(2) - a - 1 are irreducible over Q, which answers a question posed in [1], except for three values of a. Finally, we make a conjecture that suggests a necessary and sufficient condition for the stability of any monic, post-critically finite quadratic polynomial over any field of characteristic not equal 2.