The Galois group of random elements of linear groups

Abstract

Let F be a finitely generated field of characteristic zero and <GLn(F) a finitely generated subgroup. For an element g in , let Gal(F(g)/ F) be the Galois group of the splitting field of the characteristic polynomial of g over F. We show that the structure of Gal(F(g)/ F) has a typical behaviour depending on F, and on the geometry of the Zariski closure of (but not on ).

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