The Galois group of random elements of linear groups (Q2927717)

From authors' abstract: ``Let \(F\) be a finitely generated field of characteristic zero and \(\Gamma \leq \mathrm{GL} _n (F)\) a finitely generated subgroup. For \(\gamma \in \Gamma \), let \(\mathrm{Gal} (F(\gamma)/F\)) be the Galois group of the splitting field of the characteristic polynomial of \(\gamma\) over \(F\). We show that the structure of \(\mathrm{Gal} (F(\gamma)/F\)) has a typical behavior depending on \(F\) and the geometry of the Zariski closure of \(\Gamma \) (but not on \(\Gamma \)). ''NEWLINENEWLINENEWLINE The symbol \({\mathbb F}\) is used in the paper for a general field.