Let f(t,X) be an irreducible polynomial in X with coefficients in Q(t). By Hilbert's irreducibility theorem, the polynomial f(t_0,X) is still irreducible over Q for infinitely many integers t_0. We show that the set of these t_0 is even cofinite in the integers Z under several general assumptions, and give non--trivial cases where cofiniteness in Z fails. The proofs are based on Hilbert's classical reduction argument in the proof of his theorem, Siegel's theorem about integral points on algebraic curves, a connection of the Galois group of f(t,X) over Q(t) to monodromy groups of rational functions g(Z) in Q(Z) which admit infinitely many integral values on Q, and a careful analysis of these latter groups.
As a sample application, we obtain the following result: Let f(t,X) in Q(t)[X] be irreducible of degree at least 3 and with a simple Galois group which is not isomorphic to an alternating group. Then this Galois group is preserved for all but finitely many integral specializations of t.
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