$A \times B^{-1}$ has irreducible characteristic polynomial when $A,B$ are random integer matrices -- simple proof?

Question

Let $A,B$ be $d\times d$ integer matrices with each entry drawn uniformly from $[0,2^n)$, and define the rational matrix $C = A \times B^{-1}$. Is there a simple way to prove that $C$'s characteristic polynomial is irreducible over the rationals with high probability (say $1-$exp$(-nd)$)?