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arXiv · 2410.03792

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

Manjul Bhargava
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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 $n$.

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Manjul Bhargava. 2024-10-03. A proof of van der Waerden's Conjecture on random Galois groups of polynomials. https://arxiv.org/abs/2410.03792

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