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

Abstract

Of the $(2H+1{)}^n$ monic integer polynomials $f(x)=x^n+a_1{x}^{n-1}+\cdots +a_n$ with $\max \{|a_1|,\dots ,|a_n|\}\le H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as may be obtained by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees $n\le 4$, 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 $n$.

13 pages; to appear in Pure and Applied Mathematics Quarterly - Special Issue in honor of Don Zagier's 70th birthday. arXiv admin note: substantial text overlap with arXiv:2111.06507