A finiteness theorem for specializations of dynatomic polynomials

Let t and x be indeterminates, let phi(x) = x(2) + t is an element of Q (t)[x], and for every positive integer n let Phi(n) (t, x) denote the n-th dynatomic polynomial of phi. Let G(n) be the Galois group of Phi(n) over the function field Q(t), and for c is an element of Q let G(n,c) be the Galois group of the specialized polynomial Phi(n )(c, x). It follows from Hilbert's irreducibility theorem that for fixed n we have G(n) congruent to G(n,c) for every c outside a thin set E-n subset of Q. By earlier work of Morton (for n = 3) and the present author (for n = 4), it is known that E-n is infinite if n <= 4. In contrast, we show here that E-n is finite if n is an element of {5, 6, 7, 9}. As an application of this result we show that, for these values of n, the following holds with at most finitely many exceptions: for every c is an element of Q, more than 81% of prime numbers p have the property that the polynomial x(2) + c does not have a point of period n in the p-adic field Q(p).