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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 $p$ divides the discriminat of $f(X)$. Let $A_p$ be the localization of $A$ with respect to $S = \mathbb{Z} - p\mathbb{Z}$.

Are there any criteria to assure that $A_p$ is integrally closed?

Motivation Combining with this, maybe we can get criteria to assure that $A$ is integrally closed, i.e. it is the ring of algebraic integers in $K$.

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