The set of stable primes for polynomial sequences with large Galois group

Abstract

Let K be a number field with ring of integers O-K, and let {f(k)}(k is an element of N) be a sequence of monic polynomials in O-K[x] such that for every n is an element of N, the composition f((n)) = f(1) circle f(2) circle... circle f(n) is irreducible. In this paper we show that if the size of the Galois group of f((n)) is large enough (in a precise sense) as a function of n, then the set of primes p subset of O-K such that every f((n)) is irreducible modulo p has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of f((n)) is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences.