Power-saving error terms for the number of $D_4$-quartic extensions over a number field ordered by discriminant

TL;DR

This paper provides an explicit count and power-saving error term for the number of dihedral quartic extensions over a general number field, advancing understanding of their distribution ordered by discriminant norms.

Contribution

It extends previous results by deriving both the main term and a power-saving error term for $D_4$-quartic extensions over arbitrary number fields.

Findings

Explicit main term for counting $D_4$-quartic extensions.

Power-saving error term established for general base fields.

Comprehensive overview of number field asymptotics history.

Abstract

We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the literature (see Cohen--Diaz y Diaz--Olivier for the statement without proof and see Kl\"uners for a proof), but a power-saving for the error term has not been explicitly determined except in the case that the base field is $\mathbb{Q}$. In this article, we describe the argument for obtaining both the explicit main term and a power-saving error term for the number of $D_4$-quartic extensions over a general base number field ordered by the norms of their relative discriminants. We also give an extensive overview of the history and development of number field asymptotics.