Let f(X,t) in Q[X,t] be an irreducible polynomial. Hilbert's irreducibility theorem asserts that there are infinitely many t_0 in Z such that f(X,t_0) is still irreducible. We say that f(X,t) is general if the Galois group of f(X,t) over Q(t) is the symmetric group in its natural action. We show that if the degree of f with respect to X is a prime different from 5 or if f is general of degree different from 5, then f(X,t_0) is irreducible for all but finitely many t_0 in Z unless the curve given by f(X,t)=0 has infinitely many points (x_0,t_0) with x_0 in Q, t_0 in Z. The proof makes use of Siegel's theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others.
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