An improved error term for counting $D_4$-quartic fields

TL;DR

This paper improves the error term in counting $D_4$-quartic fields with bounded discriminant, providing more precise asymptotics and extending results to dihedral extensions over arbitrary base fields.

Contribution

It introduces a sharper error estimate for counting $D_4$-quartic fields and generalizes the result to dihedral extensions over any base field.

Findings

Error term improved to $O(X^{5/8+\varepsilon})$ for $D_4$-quartic fields

Asymptotic formula with explicit constant $C$ for counting such fields

Extension of results to quartic dihedral extensions over arbitrary base fields

Abstract

We prove that the number of quartic fields $K$ with discriminant $|\Delta_K|\leq X$ whose Galois closure is $D_4$ equals $CX+O(X^{5/8+\varepsilon})$, improving the error term in a well-known result of Cohen, Diaz y Diaz, and Olivier. We prove an analogous result for counting quartic dihedral extensions over an arbitrary base field.