On the distribution of Galois groups

TL;DR

This paper provides an upper bound on the number of monic integer polynomials with a given Galois group, showing that polynomials with smaller Galois groups are relatively rare among all polynomials of bounded height.

Contribution

It establishes a quantitative bound on the count of polynomials with a specified Galois group, linking group index to polynomial enumeration.

Findings

Bound of O_{n, ε}(H^{n-1+δ_G+ε}) for polynomials with Galois group G

Fewer polynomials have small Galois groups compared to the total

Quantitative relationship between Galois group size and polynomial count

Abstract

Let $G$ be a subgroup of the symmetric group $S_n$, and let $\delta_G=|S_n/G|^{-1}$ where $|S_n/G|$ is the index of $G$ in $S_n$. Then there are at most $O_{n, \epsilon}(H^{n-1+\delta_G+\epsilon})$ monic integer polynomials of degree $n$ having Galois group $G$ and height not exceeding $H$, so there are only `few' polynomials having `small' Galois group.