# Counting $G$-Extensions by Discriminant

## TL;DR

This paper develops new upper bounds for counting Galois extensions of number fields with fixed degree and bounded discriminant, using geometry of numbers and invariant theory, advancing understanding of number field distribution.

## Contribution

It introduces a novel approach combining geometry of numbers and invariant theory to bound the number of Galois extensions with fixed degree and discriminant.

## Key findings

- Provides explicit upper bounds for Galois extensions with given parameters.
- Extends previous methods to include specified Galois closure.
- Enhances understanding of the distribution of number field extensions.

## 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.