Given a number field $k$ and a quadratic extension $K_2$, we give an explicit
asymptotic formula for the number of isomorphism classes of cubic extensions of
$k$ whose Galois closure contains $K_2$ 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
$O(X^{\alpha})$, for an explicit $\alpha<1$.