# How to determine whether a given cyclotomic integer generates a prime ideal or not

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?

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One way is to compute the norm $N\alpha$ and check that it is the norm of a prime ideal, i.e. either $N\alpha=l$ or $N\alpha=p^f$, where $p\neq l$ is prime and $f$ is the smallest such that $p^f\equiv 1$ mod $l$. Notice that the norm of a non-prime ideal is not of this form (it's a product of these numbers), that's why this method works in cyclotomic fields.