The following is stated on the Wikipedia entry for integrally closed domains as an example:
Let $k$ be a field of characteristic not $2$ and $S=k[x_1,...,x_n]$ a polynomial ring over it. If $f$ is a square-free nonconstant polynomial, then $A=S[y]/(y^2-f)$ is an integrally closed domain.
If I am correct, then $A$ is just $S$ adjointed $\sqrt f $. But I still don't see how one can prove that $A$ is integrally closed.
