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 $P(x)\in \text{Zz}[x]$ to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if $f(t_1,\mathrm{.}\mathrm{.}\mathrm{.},t_r,x)$ is an indecomposable polynomial in several variables with coefficients in a field of characteristic $p=0$ or $p>\mathrm{deg}(f)$, then the one variable specialized polynomial $f(t_1^{\ast }+{\alpha }_1^{\ast }x,\mathrm{.}\mathrm{.}\mathrm{.},t_r^{\ast }+{\alpha }_r^{\ast }x,x)$ is indecomposable for all $(t_1^\ast, …, t_r^\ast, \alpha_1^\ast, …,\alpha_r^\ast)\in \bar k^{2r}$ off a proper Zariski closed subset.