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In this article we study asymptotic behavior of the probability that a randommonic polynomial with integer coefficients is irreducible over the integers. Weconsider the cases where the coefficients grow together with the degree of therandom polynomials. Our main result is a generalization of a theorem proved byKonyagin in 1999. We also generalize Hilbert's Irreducibility Theorem andpresent an analog of this result with centered Binomial distributedcoefficients.

On the probability of irreducibility of random polynomials with integer coefficients