For commutative rings $R \subseteq S$,
recall that $S$ is [separable][1] over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.

> **Question 1:** Is $\mathbb{C}[x]$ separable over $\mathbb{C}[x^2,x^3]$?

More generally,

> **Question 2:** Is it possible to characterize all $\mathbb{C}$-subalgebras $\mathbb{C} \subset R \subset \mathbb{C}[x]$ such that $\mathbb{C}[x]$ is separable over $R$?

According to wikipedia: "Moreover, an algebra $S$ is separable if and only if it is flat when considered as a right module of $S \otimes_R S$ in the usual way".
Here $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$ is not flat; I am not sure if there is a connection between flatness or non-flatness of $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$ and $\mathbb{C}[x] \otimes_{\mathbb{C}[x^2,x^3]} \mathbb{C}[x] \subset \mathbb{C}[x]$.

If, for example, $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x] \otimes_{\mathbb{C}[x^2,x^3]} \mathbb{C}[x]$ is flat (I do not know if this is true or false), then
flatness of $\mathbb{C}[x] \otimes_{\mathbb{C}[x^2,x^3]} \mathbb{C}[x] \subset \mathbb{C}[x]$ would imply flatness of $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$, which is false.

I have asked the above question [here][2].
[This][3] question may be relevant.

**Edit:** After receiving a comment that "it's unlikely that you can characterise all $R$ for which $R \subseteq \mathbb{C}[x]$ is separable, I would like to change **Question 2** to the following question:

> **Question 3:** Is it possible to characterize all $h \in \mathbb{C}[x]$ with $\deg(h) \geq 2$, such that $\mathbb{C}[x]$ is separable over:
> 
> (i) $A=\mathbb{C}[h]$. 
> 
> (ii) $B=\mathbb{C}+(h)$, where $(h)$ denotes the ideal of $\mathbb{C}[x]$ generated by $h$.


Thank you very much!

  [1]: https://en.wikipedia.org/wiki/Separable_algebra
  [2]: https://math.stackexchange.com/questions/3829835/separability-of-mathbbcx-over-its-mathbbc-subalgebras
  [3]: https://mathoverflow.net/questions/316933/maximal-subalgebras-in-polynomial-ring-mathbbrx-over-the-field-mathbbr