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

Abstract

Let $n\ge 6$ 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 $n\ge 8$ 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.

removed Lee citation