A proof of van der Waerden’s Conjecture on random Galois groups of polynomials

Of the (2H +1)ⁿ monic integer polynomials f(x) = xⁿ+a₁xⁿ⁻¹+···+a_n with max{|a₁|,…,|a_n|} ≤ H,howmanyhave associated Galois group that is not the full symmetric group S_n? There are clearly ≫ Hⁿ⁻¹ such polynomials, as may be obtained by setting a_n = 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. In this expository article, we outline a proof of van der Waerden’s Conjecture for all degrees n.^∗.