Of the (2H+1)n monic integer polynomials f(x)=xn+a1xn−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. In this talk, we will describe a proof of van der Waerden’s Conjecture for all degrees n.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
