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

TL;DR

This paper proves van der Waerden's conjecture, establishing that the number of integer polynomials with non-full symmetric Galois groups grows on the order of H^{n-1}, confirming the conjecture for all degrees.

Contribution

The paper provides a complete proof of van der Waerden's conjecture for all polynomial degrees, extending previous results limited to degrees up to 4.

Findings

Number of polynomials with non-full symmetric Galois group is O(H^{n-1})

Confirms van der Waerden's conjecture for all degrees

Extends prior degree-specific results to general case

Abstract

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 expository article, we outline a proof of van der Waerden's Conjecture for all degrees $n$.