Let $K$ be a number field containing $\mu_3$, the third roots of unity. Consider a monic irreducible cubic polynomial $f \in K[x]$ whose discriminant $\Delta$ is a square in $K$. Thus the splitting field $L$ of $f$ gives a $\mathbb{Z}/3\mathbb{Z}$-extension of $K$, and by Kummer Theory, we have
$ L = K(\sqrt[3]{a})$
for some non-cube $a \in K$. I would like to know how to find such an $a$.
Perhaps more concretely, the splitting field of $f = x^3 + Ax + B$ (say) is equal to the splitting field of an irreducible cubic of the form $x^3 - a$ (by Kummer Theory), and I guess I'm asking, how do I go from $x^3 + Ax + B$ to $x^3 - a$?