restricted access The Galois group of random elements of linear groups

Let $\Bbb {F}$ be a finitely generated field of characteristic zero and $\Gamma\leq{\rm GL}_n(\Bbb {F})$ a finitely generated subgroup. For $\gamma\in\Gamma$, let ${\rm Gal}(\Bbb {F}(\gamma)/\Bbb {F})$ be the Galois group of the splitting field of the characteristic polynomial of $\gamma$ over $\Bbb {F}$. We show that the structure of ${\rm Gal}(\Bbb {F}(\gamma)/\Bbb {F})$ has a typical behavior depending on $\Bbb {F}$, and on the geometry of the Zariski closure of $\Gamma$ (but not on $\Gamma$).