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

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