Title: Counting $G$-Extensions by Discriminant,
Abstract: The problem of analyzing the number of number field extensions $L/K$ with
bounded (relative) discriminant has been the subject of renewed interest in
recent years, with significant advances made by Schmidt, Ellenberg-Venkatesh,
Bhargava, Bhargava-Shankar-Wang, and others. In this paper, we use the geometry
of numbers and invariant theory of finite groups, in a manner similar to
Ellenberg and Venkatesh, to give an upper bound on the number of extensions
$L/K$ with fixed degree, bounded relative discriminant, and specified Galois
closure.