Is $\mathbb{C}[X, Y] / (X^5 + Y - 13)$ integrally closed?

Question

I want to check if $R := \mathbb{C}[X, Y] / (X^5 + Y - 13)$ is a Dedekind domain or not.

1. I know $R$ is an integral domain because $X^5 + Y - 13$ is prime in $\mathbb{C}[X, Y]$.
2. $R$ is also noetherian which is easy to check.
3. I think Krull dimension is 1 because $(X^5 + Y -13)$ "divides" out polynomials with two variables.

But I have no idea how to see if $R$ is integrally closed or not. My intuition is yes which would mean $R$ is a Dedekind domain.

So I don't know if it helps, but $\mathbb{C}[X, Y]$ is a UFD, but the quotient doesn't have to be. I've seen some approaches on this site, but I would like to solve this algebraically and not geometrically. Thank you in advance.

Answer 1

Note that $\mathbb{C}[X,Y]/(X^5+Y-13)\cong \mathbb{C}[X]$ via the first isomorphism theorem applied to the surjective ring homomorphism $f(X,Y)\mapsto f(X,13-X^5)$.

And $\mathbb{C}[X]$ is a UFD.