Let $K$ be a cubic extension of the rational numbers of discriminant $D$ and $\{ 1, \omega_2, \omega_3 \}$ be an integral basis for the ring of integers $\mathcal{O}_K$ of $K$. Let $\alpha \in \mathcal{O}_K$ be primitive so that no rational prime divides $\alpha$, let the norm of $\alpha$ be equal to $a^3$, with $a \in \mathbb{Z}$, and assume that $a$ is prime to $D$.

**Question: Do there always exist rational integers $\beta_2$, $\beta_3$ such that $$\{ a, \beta_2 + \omega_2, \beta_3 + \omega_3 \}$$ is an integral basis for the ideal $(a, \alpha )$ ?** Even when $\gcd (disc.(\alpha ), a ) \not= 1$ ?