Generators of a Power of a Prime Ideal

Question

Let $K = \mathbb{Q}(\zeta_n)$, where $\zeta_n$ is an nth root of unity. I know that in $\mathcal{O}_K = \mathbb{Z}[\zeta_n]$, a prime ideal above a given prime $p\in\mathbb{Z}$ has the form $\mathfrak{p} = (p, f(\zeta_n))$, where $f$ is an irreducible polynomial in the factorization of $\Phi_n(x)$ modulo $p$. I have a question about the generators of $\mathfrak{p}^2$: is $f(\zeta_n)$ contained in $\mathfrak{p}^2?$ I know that $\mathfrak{p}^2$ is generated by $p^2, f(\zeta_n)^2,$ and $pf(\zeta_n)$, but I can't seem to prove that it doesn't also contain $f(\zeta_n)$.