Irreducible polynomials of bounded height

Abstract

The goal of this paper is to prove that a random polynomial with independent and identically distributed random coefficients taking values uniformly in $\{1,\dots ,210\}$ 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 $1$.