I've been pondering the following situation for a little while now. Let $p$ be a prime number, and I denote $\zeta:=\zeta_p$ to be a primitive $p^{\text{th}}$ root of unity.
Consider the rings $\mathbb{Z}$ and $\mathbb{Z}[\zeta]$. Now $\mathbb{Z}[\zeta]$ is a finitely generated $\mathbb{Z}$ module, hence $\mathbb{Z}[\zeta]$ is integral over $\mathbb{Z}$. Of course $p\mathbb{Z}$ is a prime ideal of $\mathbb{Z}$, hence it known that there exists a prime ideal $\mathfrak{P}$ lying above $p\mathbb{Z}$, that is, $\mathfrak{P}\cap\mathbb{Z}=p\mathbb{Z}$. Moreover, $p\mathbb{Z}$ is maximal since $\mathbb{Z}/p\mathbb{Z}$ is a field, and thus $\mathfrak{P}$ is maximal in $\mathbb{Z}[\zeta]$.
Can we say something stronger though, that there exists a unique maximal ideal $\mathfrak{M}$ of $\mathbb{Z}[\zeta]$ such that $\mathfrak{M}\cap\mathbb{Z}=p\mathbb{Z}$ with the additional property that $\mathfrak{M}^{p-1}=p\mathbb{Z}[\zeta]$?
Many thanks for your explanations.