Is the ideal $x^3-y^5 \subseteq \mathbb{C}[x,y]$ prime? Is it maximal?

Question

I have some idea like I is a maximal ideal of a commutative ring R iff R/I is a field. but not able to formulate for this case. first, I thought about the irreducibility of ideal $x^3-y^5 \subseteq \mathbb{C}[x,y]$ and show if $I$ is irreducible then it is prime

Answer 1

Consider the ring homomorphism $\phi: \mathbb{C}[x,y] \to \mathbb{C}[t]$ given by $f(x,y) \mapsto f(t^5,t^3)$.

Then $\ker \phi = \langle x^3-y^5 \rangle =I$.

Now $\operatorname{im}\phi$ is a domain because it is a subring of $\mathbb{C}[t]$, which is a domain. Therefore, $I$ is prime.

On the other hand, the units of $\mathbb{C}[t]$ are $\mathbb{C}^*$ and so $\operatorname{im}\phi$ is not a field. Therefore, $I$ is not maximal.