Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

up vote 3 down vote accepted

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.

share|improve this answer
    
Thanks. Is that a sufficient condition? –  Makoto Kato Jul 23 '12 at 9:43
    
yes, as I wrote, non-prime ideals don't have these norms –  user8268 Jul 23 '12 at 9:51

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.