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

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?

• "Just" show $\;\Bbb Z[\zeta_p]\;$ is the ring of algebraic integers of the number field $\;\Bbb Q(\zeta_p)\;$ . Then it automatically follows it is a Dedekind ring. Apr 1 '17 at 13:16
• How would I go about doing that? Apr 1 '17 at 14:32
• You may want to try this: math.uiuc.edu/~r-ash/Ant/AntChapter7.pdf It is not trivial and can be lengthy to show such a thing. Apr 1 '17 at 15:54
• For the proof of unique factorization of ideals in rings of integers, the main step is en.wikipedia.org/wiki/Primary_decomposition and the fact that if $\mathcal{P}$ is a prime ideal then $mathcal{O}_K/ \mathcal{P}$ is an integral domain with finitely many elements and hence a field. Apr 1 '17 at 17:00
• The simpler example is $w =i \sqrt{5}, \mathcal{O}_K= \mathbb{Z}[w]$ whose class group has two elements : $(1)$ and $(2,1+w)$, so its ideals are of the form $(a), a \in \mathcal{O}_K$ or $(a,a\frac{1+w}{2}), a \in \mathcal{O}_K$ Apr 1 '17 at 17:16