Power-Saving Error Terms for the Number of \(D_4\) -Quartic Extensions over a Number Field Ordered by Discriminant

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 et al (Compos Math 133(1):65–93, 2002) for the statement without proof and see Klüners (Int J Number Theory 8(3):845–858, 2012) 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 chapter, 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.