MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 3 added 157 characters in body; edited tags

I am desperately looking for an integral domain $A$ with the following properties:

  1. $A$ is not integrally closed
  2. $A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
  3. 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.
show/hide this revision's text 2 added 6 characters in body

I am desperately looking for an integral domain $A$ with the following properties:

  1. $A$ is not integrally closed
  2. $A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
  3. There is an integral element $x\in K$ of (over $A$ A$) such that $A[x]$ is integrally closed.

Can someone help to tell me if the above is even possible?

show/hide this revision's text 1

normal domains with algebraically closed quotient field

I am desperately looking for an integral domain $A$ with the following properties:

  1. $A$ is not integrally closed
  2. $A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
  3. There is an integral element $x\in K$ of $A$ such that $A[x]$ is integrally closed.

Can someone help to tell me if the above is even possible?