On the probability of irreducibility of random polynomials with integer coefficients

TL;DR

This paper investigates the asymptotic probability that a random monic polynomial with integer coefficients is irreducible, extending previous theorems and considering coefficients that grow with the degree.

Contribution

It generalizes Konyagin's 1999 theorem and Hilbert's Irreducibility Theorem to broader classes of random polynomials with integer coefficients.

Findings

Probability of irreducibility approaches 1 as degree increases

Generalized theorem for coefficients growing with degree

Extended Hilbert's Irreducibility Theorem for binomial coefficients

Abstract

In this article we study asymptotic behavior of the probability that a random monic polynomial with integer coefficients is irreducible over the integers. We consider the cases where the coefficients grow together with the degree of the random polynomials. Our main result is a generalization of a theorem proved by Konyagin in 1999. We also generalize Hilbert's Irreducibility Theorem and present an analog of this result with centered Binomial distributed coefficients.