A proof of van der Waerden’s Conjecture on random Galois groups of polynomials

Of the $(2H+1)^n$ monic integer polynomials $f(x) =x^n + a_1 x^{n-1} + \dotsc + a_n$ with $\max{\lbrace \lvert a_1 \rvert, \dotsc , \lvert a_n \rvert \rbrace} \leq H$, 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 expository article, we outline a proof of van der Waerden’s Conjecture for all degrees $n$.

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Received 13 December 2022

Received revised 4 February 2023

Accepted 13 February 2023

Published 3 April 2023