Abstract
Of the $(2H+1)^n$ monic integer polynomials $f(x) = x^n +a_1x^{n-1}+\cdots a_n$ with $\max(\{|a_1| \dots,|a_n|\} \le H$, how many 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 aerden 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 $n$.
