Irreducible but not prime in $K[X,Y]/(X^2-Y^3)$

Question

I'd like to know why $X+(X^2-Y^3)$ and $Y+(X^2-Y^3)$ are irreducible, but not prime in $K[X,Y]/(X^2-Y^3)$.

I failed in both. For the first, I tried to use an isomorphism $$\phi:K[X,Y]/(X^2-Y^3)\to K[X], a_1=0$$ by $X\to X^3, Y\to X^2$ and for the second I tried to construct a case similar to the factorization $3^2=(2+\sqrt-5)(2-\sqrt-5)$.

Answer 1

Notation. Define $x := X + (X^2-Y^3)$ and $y:= Y + (X^2-Y^3)$, so that $x^2=y^3$

Hint. Clearly $x\mid y^3$ whereas $x\not\mid y$. In the same vein $y\mid x^2$ whereas $y\not\mid x$. This shows that the elements are not prime.