Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $\mathbb{Q}(\zeta)$ be the cyclotomic field. Let $A$ be the ring of algebraic integers in $\mathbb{Q}(\zeta)$. Let $\alpha$ be a non-zero element of $A$.
My question: How can we determine whether $\alpha A$ is a prime ideal or not?