Counting cubic extensions with given quadratic resolvent

Given a number field k and a quadratic extension K2, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of k whose Galois closure contains K2 as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is O(Xα), for an explicit α<1.