  Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$
with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, 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 $n\leq 4$, 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 $n$.
