  We prove that the number of unimodular integral matrices in a norm ball whose
characteristic polynomial has Galois group different than the full symmetric
group is of strictly lower order of magnitude than the number of all such
matrices in the ball, as the radius increases. More generally, we prove a
similar result for the Galois groups associated with elements in any connected
semisimple linear algebraic group defined and simple over a number field $F$.
Our method is based on the abstract large sieve method developed by Kowalski,
and the study of Galois groups via reductions modulo primes developed by Jouve,
Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice
point counting result, and a non-concentration phenomenon for lattice points in
algebraic subvarieties of the group variety, both established previously by the
authors. The results answer a question posed by Rivin and by Jouve, Kowalski
and Zywina, who have considered Galois groups of random products of elements in
algebraic groups.
