Towards van der Waerden's conjecture

TL;DR

This paper estimates the number of monic, irreducible polynomials with integer coefficients within a certain range that are solvable by radicals, confirming a 1936 conjecture of van der Waerden for most degrees.

Contribution

It provides an asymptotic bound on the count of such polynomials, advancing the understanding of van der Waerden's conjecture for degrees other than 7, 8, and 10.

Findings

Number of solvable quintic polynomials is O(H^{3.91})

For degrees n ≥ 3, n not in {7,8,10}, the count is O(H^{n-1.017})

Confirms van der Waerden's conjecture for most degrees except 7, 8, 10

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 $n \ge 3$ 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.