Prove that the normalisation of $A=k[X,Y]/(Y^2-X^2-X^3)$ is $k[t]$ where $t=Y/X$ (Reid, Exercise 4.5)

This is a problem about finding the normalisation of a quotient polynomial ring. So I have to find the integral closure of the ring in its field of fractions. The problem statement is as follows:

Let $A=k[X,Y]/(Y^2-X^2-X^3)$. Prove that the normalisation of $A$ is $k[t]$ where $t=Y/X$.

Can I do this by showing that the field of fractions $\text{Frac} A$ of $A$ is equal to $k[t]$, and subsequently showing that the field of fractions is normal? (This could be done by showing that $k[t]$ is a UFD?)

I am lost at calculating/determining $\text{Frac}A$, or similarly proving that $k[t]=\text{Frac}A$. Also, how do I show that it is normal?

I hope you can help!

Solution 1:

Hints.

• Consider $\varphi:K[X,Y]\to K[T]$ given by $\varphi(X)=T^2-1$, $\varphi(Y)=T(T^2-1)$. Prove that $\ker\varphi=(Y^2-X^2-X^3)$.
• We also have $A\simeq\operatorname{Im}\varphi=K[T^2-1,T(T^2-1)]\subset K[T]$, and $T$ is integral over $K[T^2-1,T(T^2-1)]$.