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A proof of van der Waerden's Conjecture on random Galois groups of polynomials (2410.03792v2)

Published 3 Oct 2024 in math.NT

Abstract: Of the $(2H+1)n$ monic integer polynomials $f(x)=xn+a_1 x{n-1}+\cdots+a_n$ 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 $n$.

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