Joint IAS/Princeton University Number Theory Seminar

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.  In this talk, we will describe a proof of van der Waerden's Conjecture for all degrees $n$.