On the distribution of Galois groups

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.