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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.

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  • $\begingroup$ Thanks. Is that a sufficient condition? $\endgroup$ – Makoto Kato Jul 23 '12 at 9:43
  • $\begingroup$ yes, as I wrote, non-prime ideals don't have these norms $\endgroup$ – user8268 Jul 23 '12 at 9:51

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