Specializations of indecomposable polynomials

We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime p for the reduction modulo p of an indecomposable polynomial to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if f ( t 1 , . . . , t r , x ) is an indecomposable polynomial in several variables with coefficients in a field of characteristic p = 0 or p > deg( f ), then the one variable specialized polynomial is indecomposable for all outside a proper Zariski closed subset.