Finiteness results for Hilbert's irreducibility theorem

Let k be a number field, O-k its ring of integers, and f(t, X) E k(t) [X] be an irreducible polynomial. Hilbert's irreducibility theorem gives infinitely many integral specializations t --> (t) over bar is an element of O-k such that f((t) over bar, X) is still irreducible. In this paper we study the set Red(f) (O-k) of those (t) over bar is an element of O-k with f ((t) over bar, X) reducible. We show that Red(f)(O-k) 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 theory, valuation theory, and Siegel's theorem about integral points on algebraic curves. Indeed, using the Siegel-Lang extension of Siegel's theorem, most of our results hold over more general fields. Using the classification of the finite simple groups, further results can be obtained. The last section contains a short survey.