Towards van der Waerden’s conjecture

Chow, Sam; Dietmann, Rainer

doi:10.1090/tran/8784

Towards van der Waerden’s conjecture
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by Sam Chow and Rainer Dietmann;

Trans. Amer. Math. Soc. 376 (2023), 2739-2785

DOI: https://doi.org/10.1090/tran/8784

Published electronically: January 24, 2023

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Abstract:

How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in $[-H,H]$, is $O(H^{3.91})$. More generally, we show that if $n \geqslant 3$ and $n \notin \{ 7, 8, 10 \}$ then there are $O(H^{n-1.017})$ monic, irreducible polynomials of degree $n$ with integer coefficients in $[-H,H]$ and Galois group not containing $A_n$. Save for the alternating group and degrees $7,8,10$, this establishes a 1936 conjecture of van der Waerden.

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