Ideals of $\mathbb{Z}[\zeta_{p}]$ factorise uniquely

Question

I am trying to show that the ideals of $\mathbb{Z}[\zeta_{p}]$ factorise uniquely.

In know that $\mathbb{Z}[\zeta_{p}]$ is not a UFD in general. I also know that, for Dedekind rings, non-zero proper ideals have unique factorisation as a product of non-zero prime ideals. I think I just need to show that $\mathbb{Z}[\zeta_{p}]$ is a Dedekind ring.

Is that the case? If so, how to I show it? If not, what do I need to show?

Answer 1

For the proof of unique factorization of ideals in ring of integers, I think the main steps are :

• $\mathbb{Z}[X_1,\ldots,X_m]$ is Noetherian so that $\mathcal{O}_K= \mathbb{Z}[X_1,\ldots,X_m]/I$ is Noetherian

• The primary decomposition in Noetherian rings

• If $\mathcal{P}$ is a prime ideal then $\mathcal{O}_K/ \mathcal{P}$ is an integral domain with finitely many elements, hence it is a field.

• Projecting in those finite fields, it follows that the primary ideals are powers of prime ideals and the decomposition is unique