Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $p$ be a prime number. Suppose $pA = \alpha^n A$, where $\alpha \in A$ and $n = [K : \mathbb{Q}]$. Let $A_p$ be the localization of $A$ with respect to $S = \mathbb{Z} - p\mathbb{Z}$.
My question: Is $A_p$ integrally closed?
Motivation Let $p$ be an odd prime number. Let $\theta$ be a $p$-th primitive root of unity. Let $\alpha = 1 - \theta$. Then it is well known that $pA = \alpha^n A$. It is also well known that $A_p$ is integrally closed.
This is a related question.