what is the unique prime factorization for the ideal $p\mathbb{Z}[\zeta]$ in the Dedekind domain $\mathbb{Z}[\zeta]$?

Question

Let $p$ be a prime number and $\zeta=e^{\frac{2\pi}{p}i}$. I want to find the unique factorization into a product of prime $\mathbb{Z}[\zeta]$-ideals for the ideal $p\mathbb{Z}[\zeta]$. Now, I know that for some invertible element $a\in \mathbb{Z}[\zeta]$, I can write $p=a(1-\zeta)^{p-1}$. I also know that the ideal $(1-\zeta)$ is prime. Then, isn't it a direct consequence that $$p\mathbb{Z}[\zeta]=(1-\zeta)^{p-1}$$ is the unique prime factorization for $p\mathbb{Z}[\zeta]$? If it is not correct, then please point my mistake. If it is correct, do I need to prove anything else for my proof to be complete?