Let $k$ be a number field, ${𝒪}_k$ its ring of integers, and $f(t,X)\in k\left(t\right)\left[X\right]$ be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations $t\mapsto \overline{t}\in {𝒪}_k$ such that $f(\overline{t},X)$ is still irreducible. In this paper we study the set ${\mathrm{Red}}_f\left({𝒪}_k\right)$ of those $\overline{t}\in {𝒪}_k$ with $f(\overline{t},X)$ reducible. We show that ${\mathrm{Red}}_f\left({𝒪}_k\right)$ is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group...