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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?

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    Why do you think Serre's criterion not useful? What have you tried? – 2017-01-26
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    Actually, for checking whether a one-dimensional domain is normal, there is nothing more useful than Serre's criterion, since it tells you that normal is equivalent to regular in this case. – 2017-01-26
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    I agree with the previous two comments. The Jacobian criterion will show everything one needs. – 2017-01-26

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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.