Abstract

We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group $G$, an affine variety $V$ and a finite map $\pi :V\rightarrow G$, all defined over a finitely generated field $\kappa $ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $\pi )$ contains a Zariski dense sub-semigroup $\Gamma \subset G$; namely, there must exist an unramified covering $p:\tilde{G}\rightarrow G$ and a map $\theta :\tilde{G}\rightarrow V$ such that $\pi \circ \theta =p$. In the case $\kappa =\mathbb{Q}$, $G=\mathbb{G}_{a}$ is the additive group, we reobtain the original Hilbert Irreducibility Theorem. Our proof uses a new diophantine result, due to Ferretti and Zannier [9]. As a first application, we obtain a necessary condition for the existence of rational fixed points for all the elements of a Zariski-dense sub-semigroup of a linear group acting morphically on an algebraic variety. A second application concerns the characterisation of algebraic subgroups of $\operatorname{GL}_N$ admitting a Zariski-dense sub-semigroup formed by matrices with at least one rational eigenvalue