Finiteness Results for Hilbert's Irreducibility Theorem, by Peter Mueller

Let $k$ be a number field, $O$ its ring of integers, and $f(t,X)$ an irreducible polynomial in $k(t)[X]$. Hilbert's irreducibility theorem gives infinitely many integral specializations of $t$ to values $a$ in $O$ such that $f(a,X)$ is still irreducible. In this paper we study the set $Red(O)$ of those $a$ in $O$ with $f(a,X)$ reducible. We show that $Red(O)$ is a finite set under rather weak assumptions. In particular, several results of K. Langmann, obtained by Diophantine approximation techniques, appear as special cases of some of our results. Our method is completely 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.

Peter Mueller <Peter.Mueller@iwr.uni-heidelberg.de>