Integrally closed domain

Remarks

Example: $\Z$ is integrally closed. If $p/q$ in lowest terms is a root of $x^n + a_{n-1} x^{n-1} + \cdots + a_0$ with $a_i \in \Z$, then $p^n = -q(a_{n-1} p^{n-1} + \cdots + a_0 q^{n-1})$, so $q$ divides $p^n$, which forces $q = \pm 1$. The same argument shows that every unique factorisation domain is integrally closed.

Non-example: $\Z[\sqrt{-3}]$ is not integrally closed. The element $(1 + \sqrt{-3})/2$ of its field of fractions is a root of $x^2 - x + 1$, but it is not in $\Z[\sqrt{-3}]$.

An integrally closed domain is also called a normal domain. A domain is integrally closed if and only if all its localizations at prime ideals are integrally closed.