This is an exercise in Artin's Algebra (10.3.11)
$\phi :\mathbb{C}[x,y] \rightarrow \mathbb{C}[t]$ is defined by $\phi(x)=t^2$ and $\phi(y)=t^3$
Prove the $\operatorname{Ker} \phi =$ the principal ideal generated by $y^2-x^3$
and describe the $\operatorname{Im} \phi$ explicitly
I am able to show the Ideal generated by y^2-x^3$ is contained in the kernel but don't know how to go about the other containment or show it is a principal ideal.