Let $R:=\mathbb{C}[x, y]/(y^2-x^3+x)$. I want to determine if $R$ is a normal ring.
The field of fractions of $R$ is $K=\mathbb{C}(x)[y]/(y^2-x^3+x)$. I think $R$ is normal, so I want to show that $R$ is integrally closed in $K$. I've noted that $R$ is integral over $\mathbb{C}[x]$, so $R$ is normal iff the integral closure of $\mathbb{C}[x]$ in $K$ is $R$, but this is not very useful. I've also tried to use Serre's criterion for normality, but this is not very useful too.
Any other ideas?
Let $f=y^2-x^3+x$. The conditions $0= \partial_x f =1-3x^2$ and $0=\partial_y f = 2y$ imply $y=0$ and $x^2 =\tfrac13$. However, $f$ does not vanish at either of these two points. Therefore, the curve defined by $f$ is nonsingular and in particular, it is normal.