Is the ring $\mathbb{C}[t^2,t^3]$ integrally closed?

Question

I am trying to understand if the ring $\mathbb{C}[t^2,t^3]$ in integrally closed (into its field of fractions), but I have no idea about how to proceed. All I have tried until now has failed. Any hints/ideas?

Answer 1

The field of fractions is $\mathbb{C}(t)$. The element $t$ is integral because it is a root of $x^2-t^2$. But $t \notin \mathbb{C}[t^2,t^3]$. In fact, the integral closure is $\mathbb{C}[t]$.