How to prove $ x^2 + y^3$ irreducible over $\mathbb{C}[x,y]$

Question

I am given a remark that "every irreducible polynomial in $\mathbb{C}[x]$ is exactly of degree 1",

Answer 1

It is irreducible in $(\mathbb C(y))[x]$, since it is a quadric without a root over a field.

As a polynomial in $(\mathbb C[y])[x]$ it is primitive (even monic), thus it is irreducible in that ring by Gauß' Lemma.