Irreducible polynomials of bounded height

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 $1$.