I would appreciate help with the following.
Let $k$ be a field that's algebraically closed and let $f$ be a polynomial in $k[X,Y]$. Prove that $R=k[X,Y]/(f)$ is a Dedekind domain if and only if at one of $f(a,b)$ and the two partials at $(a,b) \in k^{2}$ is not zero, for all $(a,b)$ in $k^{2}$.
Thank you all