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

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. The purpose of this paper is to prove van der Waerden's Conjecture for all degrees $n$.

34 pages; a number of clarifications and details have been added in this version; to appear in Annals of Math