Integrally Closed is Local Property

Theorem
Let $A$ be an integral domain.

For a prime ideal $\mathfrak p$ of $A$, let $A_{\mathfrak p}$ denote the localization at $S = A \backslash \mathfrak p$.

Then the following are equivalent:


 * $(1): \quad A$ is integrally closed


 * $(2): \quad A_{\mathfrak p}$ is integrally closed for all prime ideals $\mathfrak p$.


 * $(3): \quad A_{\mathfrak m}$ is integrally closed for all maximal ideals $\mathfrak m$.

$(1)$ implies $(2)$
Let $Q \left({R}\right)$ denote the quotient field of an integral domain $R$.

We have by Localization Preserves Integral Closure that:
 * $Q \left({A_{\mathfrak p} }\right) = Q \left({A}\right)$

Hence $A_{\mathfrak p}$ is integrally closed for all prime ideals $\mathfrak p$.

$(2)$ implies $(3)$
This is true because a Maximal Ideal is Prime.