$K[X^2,X^3]$ is a non-unique factorization ring

Question

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 ?

Answer 1

The $K$-algebra $K[X^2,X^3]$ is finitely generated over $K$, so by Hilbert's basis theorem, it is noetherian. In particular, it satisfies ACC on principal ideals, hence every element has a factorization into irreducibles.

Answer 2

Another approach: unique factorization domains are normal, that is, integrally closed in their field of fractions. But the field of fractions of your domain is $k(x)$, and the closure of your domain in this is $k[x]$.