I am looking for an integral domain $A$ with the following properties:
- $A$ is not integrally closed
- $A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
- There is an integral element $x\in K$ (over $A$) such that $A[x]$ is integrally closed.
Can someone help to tell me if the above is even possible?
Edit: Lubin easily gave me an example. Now I want to consider the case when I replace the condition 2. by:
2'. $A$ has a quotient field $K$ that is real closed.