Let $k$ be a field of characteristic zero, and let $t$ be a variable over $k$.
Clearly, $k[t^2,t^3] \subsetneq k[t]$ is not a UFD, since $t^2t^2t^2=t^3t^3$.
Let $h(t) \in k[t]$ with $\deg_t(h) \geq 2$.
Is it possible to somehow characterize all such $h$'s such that $k[t^2,t^3][h]$ is a UFD? (Are there such $h$'s at all?).
Any hints and comments are welcome!
Since the vector space $k[t]/k[t^2,t^3]$ is $1$-dimensional (spanned by $t$), it has only two subspaces, so there are only two possible values of $k[t^2,t^3][h]$: either $k[t^2,t^3]$ or $k[t]$. Explicitly, $k[t^2,t^3][h]=k[t]$ (and is thus a UFD) if $h$ has a linear term and $k[t^2,t^3][h]=k[t^2,t^3]$ if it does not.