Galois groups of random integer matrices

We study the number $M_n(T)$ be the number of integer $n\times n$ matrices $A$ with entries bounded in absolute value by $T$ such that the Galois group of characteristic polynomial of $A$ is not the full symmetric group $S_n$. One knows $M_n(T)\gg T^{n^2-n+1}\log T$, which we conjecture is sharp. We first use the large sieve to get $M_n(T)\ll T^{n^2-1\mathrm{/}2}\log T$. Using Fourier analysis and the geometric sieve, as in Bhargava’s proof of van der Waerden’s conjecture, we improve this bound for some classes of $A$.