OExercise I am trying to solve:
Show that the ring $R = \mathbb{C}[x^{3}, x^{2}y, xy^{2}, y^{3}]$ is normal, but $R$ is not a unique factorisation domain.
Normal means that $R$ is integrally closed in its field of fractions, i.e., $$R= \{ s \in \text{Frac}(R):s \text{ is integral over }R\},$$ where the field of fractions $\mathrm{Frac}(R)=\{\frac{f(x,y)}{g(x,y)}:f,g \in R \text{ s.t. } g\neq0\}$ and $s$ is integral over $R$ if there is a monic $h(t)\in (\mathbb{C}[x^{3}, x^{2}y, xy^{2}, y^{3}])[t]$ s.t. $h(t)=0$.
I know how to show it is not a UFD: $x^{3}y^{3}=(x^{3})(y^{3})=(x^{2}y)(xy^{2})$.
I have tried showing that any $h(x,y) \in \mathrm{Frac}(R)$ which is integral is actually in $R$, but I am not getting anywhere with it because all the ways I know to show a ring is normal use the fact that $R$ is a UFD.
Any help would be appreciated, thank you!