  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.
