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The Galois group of random elements of linear groups
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 5, October 2014
- pp. 1347-1383
- 10.1353/ajm.2014.0038
- Article
- Additional Information
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 $\Gamma$).