Suppose that it is not easy to determine that $A$ is a UFD (or that it is a local, noetherian dimension 1 domain with principal maximal ideal).
Can someone suggest strategies for showing that a finitely generated $k$-algebra is integrally closed (or not?)
I know that integral closure is a local property.
(Context: I am learning some algebraic geometry, and I need to deal with questions involving the normality of some variety. Some of the cases I can deal with using the commutative algebra that I know, but othertimes I wonder if there is some theorem that I am not aware of. I don't want to give the specific rings that I am stuck on because I would like to solve them on my own.)
Edit: $k$ is algebraically closed.