[[Integral element]]
# Integrally closed domain

An [[integral domain]] $R$ with [[field of fractions]] $K = \opn{Frac}(R)$ is **integrally closed** iff $\alpha \in K$ is integral over $R$ iff $\alpha \in R$. #m/def/ring
This motivates the **integral closure**
$$
\begin{align*}
\overline{R} = \mathcal{O}_{\opn{Frac}(R) : R}
\end{align*}
$$
which in this case is the [[Algebraic integer|ring of integers]] of the [[field of fractions]] $K$.
Thus $R$ is integrally closed iff it equals its integral closure.

## See also

- [[Dedekind domain]], which is defined to be integrally closed.

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