Upper bounds for the number of number fields with prescribed Galois group

TL;DR

This paper establishes an asymptotic upper bound on the number of degree-n number fields with a fixed Galois group and bounded discriminant, conditional on certain polynomial determinants not vanishing.

Contribution

It provides a new conditional upper bound for counting number fields with prescribed Galois groups, based on non-vanishing conditions of polynomial determinants.

Findings

Derived asymptotic upper bounds for $N_n(X;G)$ as $X$ grows large

Identified a non-vanishing condition on polynomial determinants that influences the bounds

Illustrated how to compute these determinants for specific groups

Abstract

Let $n$ be a positive integer and $G$ be a transitive permutation subgroup of $S_n$. Given a number field $K$ with $[K:\mathbb{Q}]=n$, we let $\widetilde{K}$ be its Galois closure over $\mathbb{Q}$ and refer to $Gal(\widetilde{K}/\mathbb{Q})$ as its Galois group. We may identify this Galois group with a transitive subgroup of $S_n$. Given a real number $X>0$, we set $N_{n}(X;G)$ to be the number of such number fields $K$ for which the absolute discriminant is bounded above by $X$, and for which $Gal(\widetilde{K}/\mathbb{Q})$ is isomorphic to $G$ as a permutation subgroup of $S_n$. We prove an asymptotic upper bound for $N_n(X;G)$ as $X\rightarrow\infty$. This result is conditional and based upon the non-vanishing of certain polynomial determinants in $n$-variables. We expect that these determinants are non-vanishing for many groups, and demonstrate through some examples how they may be computed.