Every non-zero prime ideal of $K[x,y]/(y^2-x^3)$ is maximal

Question

Let $R=K[x,y]/(f)$ where $f(x,y)=y^2-x^3$. I can show that $R$ is an integral domain and Noetherian. I have to show that every non-zero prime ideal of $R$ is maximal, but I can not realize the form of ideals of $R$.

I try to choose a prime ideal $P$ and want to show $R/P$ is (finite integral domain so is field) or any element in $R/P$ has inverses.

Any other suggestion. and please write me form of element such a ring. and I dont know what is identity of $R/P$

thanks a lot

Answer 1

$K[X,Y]/(Y^2-X^3)\simeq K[T^2,T^3]$, and the extension $K[T^2,T^3]\subset K[T]$ is integral, so $\dim K[T^2,T^3]=\dim K[T]=1$.