The characteristic polynomial of a random matrix

Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $ϕ$ be the characteristic polynomial. Conditionally on the extended Riemann hypothesis, with high probability $ϕ$ is irreducible and $\mathrm{Gal}(ϕ) \geq A_n$.

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