Let $\displaystyle \mathbb{K}$ be a field, consider the ring $\displaystyle \mathbb{K}[x,y]$.

1) Let $\displaystyle p(x,y) \in \mathbb{K}[x,y]$, prove that there exist $\displaystyle q(x,y) \in \mathbb{K}[x,y]$ and $\displaystyle a(x), b(x) \in \mathbb{K}[x]$ unique such that $\displaystyle p(x,y)=(x^3-y^2)q(x,y)+a(x)+b(x)y$.

2)Prove that the map $\displaystyle \phi : \mathbb{K}[x,y]\longrightarrow \mathbb{K}[t]$ defined by $\displaystyle \phi(p(x,y)) = p(t^2,t^3)$ is a ring morphism.

3)Find the kernel of $\displaystyle \phi$ (i.e: find generators of it)

4) Find explicitly the image of $\displaystyle \phi$

5)Prove that $\displaystyle x^3-y^2$ is irreducible in $\displaystyle \mathbb{K}[x,y]$ . Deduce that the image of $\displaystyle \phi$ is a domain.

My views:

1) No clue. Iīm thinking that it has to do with some division but I donīt know what...

2) Ugly computation, I used the identity given in part 1) and I think I pulled it off (if thereīs a simpler way of doing it please let me know).

3) It should be $\displaystyle <x^3-y^2>$ I suppose... Am I right?

4) $\displaystyle p(t^2,t^3) = a(t^2)+t^3b(t^2)$ means something??

5) Maybe I can use that $\displaystyle \frac{\mathbb{K}[x,y]}{<x^3-y^2>} \cong Im\phi$ once I proved that $\displaystyle x^3-y^2$ is irreducible, but I donīt know how (Eisenstein doesnīt work, I think).

Thanks, guys!!