Improved lower bounds for the number of fields with alternating Galois group

TL;DR

This paper establishes new lower bounds on the number of degree n number fields with Galois group A_n and bounded discriminant, improving previous results for larger n.

Contribution

It provides improved asymptotic lower bounds for the count of fields with Galois group A_n, advancing understanding of their distribution.

Findings

Lower bound of X^{1/8 + O(1/n)} for fields with Galois group A_n

Improved bounds for n ≥ 8 over previous results

Asymptotic growth rate of such fields with respect to discriminant

Abstract

Let $n \geq 6$ be an integer. We prove that the number of number fields with Galois group $A_n$ and absolute discriminant at most $X$ is asymptotically at least $X^{1/8 + O(1/n)}$. For $n \geq 8$ this improves upon the previously best known lower bound of $X^{(1 - \frac{2}{n!})/(4n - 4) - \epsilon}$, due to Pierce, Turnage-Butterbaugh, and Wood.