The Galois group of random elements of linear groups

TL;DR

This paper investigates the typical structure of Galois groups of elements in finitely generated linear groups over characteristic zero fields, revealing dependence on the base field and the Zariski closure's geometry.

Contribution

It establishes the typical behavior of Galois groups of elements in finitely generated linear groups, highlighting dependence on the base field and Zariski closure geometry.

Findings

Galois group structure depends on the base field

The geometry of the Zariski closure influences Galois groups

Behavior is typical across elements in the subgroup

Abstract

Let F be a finitely generated field of characteristic zero and \Gamma<GL_n(F) a finitely generated subgroup. For an element g in \Gamma, 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 \Gamma (but not on \Gamma).