Irreducible polynomials of bounded height

TL;DR

This paper demonstrates that as polynomial degree increases, a randomly chosen polynomial with coefficients uniformly in {1,...,210} is almost surely irreducible and has a Galois group containing the alternating group.

Contribution

It proves that random polynomials with bounded coefficients are almost surely irreducible and have large Galois groups as degree grows.

Findings

Probability of irreducibility tends to 1 with increasing degree.

Galois group contains the alternating group with high probability.

Results hold for coefficients uniformly in {1,...,210}.

Abstract

The goal of this paper is to prove that a random polynomial with i.i.d. random coefficients taking values uniformly in $\{1,\ldots, 210\}$ is irreducible with probability tending to $1$ as the degree tends to infinity. Moreover, we prove that the Galois group of the random polynomial contains the alternating group, again with probability tending to $1$.