Galois groups of random integer polynomials
Galois groups of random integer polynomials

Manjul Bhargava, Princeton University
Fine Hall 214
InPerson and Online Talk
Zoom Link: https://theias.zoom.us/j/88393312988?pwd=emtLbTJ5ZnMvS3hBVmNmYjhIUEFIdz09
Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n1}+\cdots+a_n$ 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 $\gg H^{n1}$ such polynomials, as may be obtained by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n1})$ 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 $n$.