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

Of the (2H+1)ⁿ monic integer polynomials f(x)=xⁿ+a₁ xⁿ⁻¹+⋯+aₙ with max\|a₁|,…,|aₙ|\≤ H, 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.