Why is $\mathbb{C}[x,y]/(y^2 - x^3 + 1)$ normal?

A problem on an algebra qual reads

Show that the ring $R = \mathbb{C}[x,y]/(y^2 - x^3 +1)$ is a Dedekind domain. (Hint: compare $R$ with the subring $\mathbb{C}[x]$.)

$R$ is clearly Noetherian. It is an integral extension of $\mathbb{C}[x]$, so inherits its dimension, which is one. How do I know it is normal?

Solution 1:

Say $A$ is an UFD containing $\frac{1}{2}$ and $d\in A$ is a square-free element, not zero, not a unit. Then the ring $B=A[y]/(y^2 - d)$ is a normal domain. Indeed, the polynomial $y^2-d$ is irreducible. The field of fractions of $B$, denoted by $L$, is $K[y]/(y^2-d)$. Consider an element $l =k_1 + k_2 \sqrt{d} \in L$. Assume $l$ is an integer. Then the trace and the norm are in $A$, that is $2k_1$ and $k_1^2 - d k_2^2 \in A$, and this implies right away $k_1$, $k_2 \in A$.