Irreducibility of random polynomials: general measures

Lior Bary-Soroker, Dimitris Koukoulopoulos, Gady Kozma

Published 2020-07-29Version 1

In this paper we prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random according to measures $\mu_0,\mu_1,\dots,\mu_{n-1}$ whose support is sufficiently large, then $f(x)$ is irreducible with probability tending to $1$ as $n$ tends to infinity. In particular, we prove that if $f(x)$ is a randomly chosen polynomial of degree $n$ and coefficients in $\{1,2,\dots,H\}$ with $H\ge35$, then it is irreducible with probability tending to 1 as $n$ tends to infinity. More generally, we prove that if we choose the coefficients of $f(x)$ independently and uniformly at random from a set $\mathcal{N}\subset[1,H]$ of $\ge H^{4/5}(\log H)^2$ integers with $H$ sufficiently large, then $f(x)$ is irreducible with probability tending to $1$ as $n$ tends to infinity. In addition, in all of these settings, we show that the Galois group of $f(x)$ is either $\mathcal{A}_n$ or $\mathcal{S}_n$ with high probability.