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, Question 3:** Assume that $S=R[w]$ is separable over $R$. Is there an additional condition that guarantees that $S$ is flat over $R$ (hence etale)?
I guess that there are examples where such $S$ is not flat over $R$.

**Remark**: If I am not wrong, if $R$ is a UFD and $w \in S$ is algebraic over
$R$ with coefficients of its minimal polynomial generating all $R$,
then $R \subseteq S=R[w]$ is flat, regardless of being separable or not.

**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 restrict 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 $R$, where: (i) $R=\mathbb{C}[h]$. (ii) $R=\mathbb{C}+(h)$, $(h)$ 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