Let $K$ be any field. I would like to prove that any element of $K[X^2,X^3]$ can be written as a product of irreducible elements, in a possibly non-unique fashion. The non-unique part can be easily proven when noticing that $$X^6 = (X^2)^3=(X^3)^2$$ and given that $X^2$ and $X^3$ are both irreducible in $K[X^2,X^3]$ (writing any of them as a product of two non-invertible elements would lie a factor of degree $1$, which can not be an element of $K[X^2,X^3]$).
However, I can't find how to prove the existence of such a decomposition. Could someone please give a hand for this exercise ?