The characteristic polynomial of a random matrix

TL;DR

This paper investigates the algebraic properties of the characteristic polynomial of random matrices with entries from b1 1, showing under the extended Riemann hypothesis that such polynomials are typically irreducible with large Galois groups.

Contribution

It establishes probabilistic irreducibility and Galois group size results for characteristic polynomials of random matrices under the extended Riemann hypothesis.

Findings

Characteristic polynomial is irreducible with high probability.

Galois group of the polynomial is at least the alternating group.

Results depend on the extended Riemann hypothesis.

Abstract

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