I was told that
$x^3-y^2$ is irreducible in $\Bbb C[x,y]$.
But I cannot really give a sounding argument. I supposed that it may be factored as $g_1, g_2$, and considered those monomials divisible by $y$. But possible cancellations makes it hard.
Suppose $(x^3-y^2)$ is reducible in $k[y][x] \cong k[x,y]$, then it would factorize as a polynomial in $x$ over the ring $k[y]$. As the degree is $3$, there must exists a monomial factorization. Hence there is a polynomial solution $x=P(y)$, which is impossible.