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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?

user72870
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    Why do you think Serre's criterion not useful? What have you tried? – Mohan Jan 26 '17 at 01:36
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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. – MooS Jan 26 '17 at 07:50
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    I agree with the previous two comments. The Jacobian criterion will show everything one needs. – Youngsu Jan 26 '17 at 08:58

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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.

Jesko Hüttenhain
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  • Please, what if instead of $R=\mathbb{C}[x, y]/(y^2-x^3+x)$, we take $S_1=\mathbb{C}[x,y]/(y^2-x^3+x,P_1)$ or more generally $S_m=\mathbb{C}[x,y]/(P_1,\ldots,P_m)$, where $m \geq 2$, $P_j \in \mathbb{C}[x,y]$. Could you please explain what Serre's criterion for normality says? I guess it involves the Jacobian matrix of $P_1,\ldots,P_m$ and its rank; the answer to the following question explains what this criterion is, however it would be nice to have an explicit example https://math.stackexchange.com/questions/69566/when-is-an-affine-algebra-normal?noredirect=1&lq=1). thank you. – user237522 Mar 24 '21 at 00:10
  • For example, $P_1=y^2-x^3+x$, $P_2=3x^2-1$ or $P_2=y^2$. – user237522 Mar 24 '21 at 00:21