Galois groups of random integer polynomials and van der Waerden’s Conjecture

Of the (2H + 1)ⁿ monic integer polynomials (Formula presented) with (Formula presented), how many have associated Galois group that is not the full symmetric group Sn? There are clearly ≫ Hⁿ⁻¹ such polynomials, as may be obtained by setting an = 0. In 1936, van der Waerden conjectured that O(Hⁿ⁻¹) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n ≤ 4, due to work of van der Waerden and Chow and Dietmann. The purpose of this paper is to prove van der Waerden’s Conjecture for all degrees n.