In 2015, I. Rivin introduced an effective method to bound the number of irreducible integral polynomials with fixed degree d and height at most N. In this paper, we give a brief summary of this result and discuss the precision of Rivin's arguments for special classes of polynomials. We also give elementary proofs of classic results on Galois groups of cubic trinomials.
Hao, Hanson; Navarro, Eli; and Stern, Henri
"Irreducibility and Galois Groups of Random Polynomials,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 22:
1, Article 10.
Available at: https://scholar.rose-hulman.edu/rhumj/vol22/iss1/10