How to prove that $R[t] = \mathbb{C}[t]$ for this integral domain?

Question

Consider the domain $$R = \frac{\mathbb{C}[X,Y]}{ (Y^2 - X^2 (X+1))} = \mathbb{C}[x,y]. $$ Let $t = y/x$ be an element in the fraction field of $R$. I have already proven that $t \notin R$. However, how can I prove that $R[t] = \mathbb{C}[t]$?

Answer 1

Well, $y^2=x^2(x+1)$ by definition of $R$, so $x+1=t^2$, and $x=t^2-1$. Hence, $y=tx=t(t^2-1)$. Thus, $R[t]=\mathbb{C}[x,y,t]=\mathbb{C}[t^2-1,t(t^2-1),t]=\mathbb{C}[t]$.