Is an integral extension of an integrally closed domain integrally closed? [closed]

Let $R$ be an integrally closed domain and $A$ an integral extension of $R$.

Is $A$ integrally closed?

Solution 1:

No. It need not be.

Consider $A=\mathbb Z\left[\sqrt{-3}\right]=\left\{a+b\sqrt{-3}\mid a,b\in\mathbb Z\right\}$ and $R=\mathbb Z$.

• $\mathbb Z$ is integrally closed.
• $\mathbb Z\left[\sqrt{-3}\right]$ is an integral extension of $\mathbb Z$ because it is isomorphic to $\mathbb Z[X]/(X^2+3)$ and $R[X]/(f)$ is always an integral extension of $R$ for a monic polynomial $f\in R[X]$.
• $\mathbb Z\left[\sqrt{-3}\right]$ is not integrally closed. Take $a=\frac{1+\sqrt{-3}}2$ in its fraction field. $a$ is integral over $\mathbb Z\left[\sqrt{-3}\right]$ because it is a root of the monic polynomial $$ X^2+X+1, $$ but $a\not\in\mathbb Z\left[\sqrt{-3}\right]$.