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.

[This][2] 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 $h \in \mathbb{C}[x]$, 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 $h$.

**Example:** If $h=x^2$, then $B=\mathbb{C}+(x^2)=\mathbb{C}[x^2,x^3] \subseteq \mathbb{C}[x]$ is not separable (first comment below).

**Remark:**
Denote $h(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0$.

According to [Corollary 8][3] with $h(Z)=c_nZ^n+c_{n-1}Z^{n-1}+\cdots+c_1Z+c_0-h$, we obtain an answer to **Question 3(i)**: $\mathbb{C}[x]$ is separable over $\mathbb{C}[h]$ iff $\deg(h)=1$ (namely, $A=\mathbb{C}[x]$).

Still, I am not sure what can be said about **Question 3(ii);** any help is welcome!

Thank you very much!


  [1]: https://en.wikipedia.org/wiki/Separable_algebra
  [2]: https://mathoverflow.net/questions/316933/maximal-subalgebras-in-polynomial-ring-mathbbrx-over-the-field-mathbbr
  [3]: https://pdf.sciencedirectassets.com/272332/1-s2.0-S0021869380X80025/1-s2.0-0021869380902331/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEAUaCXVzLWVhc3QtMSJHMEUCIQDjm2Fqy725SiofuzGVJP4E3G%2FXJjmL5QZYnjMqZZQ%2BngIgHvRHaCCWv2Ui%2F756H8sS%2BcspRvknkqHwuFpEzPXSn4wqvQMI3v%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgwwNTkwMDM1NDY4NjUiDFFXR7yWAyjQFZFKZCqRA4CdnJpaE8ptww%2Be4b1zH4vmjQTihmg2a5fC5lwbEUv40uccQgS3ZX%2FPOfiKCHnCTz0LwNBl5zv3kG23a8UhQYmcA3ZptKgzdM8eMhg7KldR84XKyvRXgdNgbwIr6I6aZ7DBxP99h7BVRFZmvREhNchfjdHJ8uV0kvE9mB9knm6R8ugIunekOx9v2tH%2BAWpUzwtmbx0fsGHtzoY2tlAncO2T3schB1pFx5GRVq9e20Cv9hDW71fGhUztVZIb8AemGp5IbrmwIKVnWhGOOGTLdm0aSlV%2Bsh1je%2FqKYruVf3OXyYJD8QXsfOjA8aWdWa3%2FWT%2BOFFmHcskv3%2Fzm80SVzbr08pWVYT02yJR6GQxXceBX9zlejip31BbNvHWB8oFVKIat821Tp3epZZ1aErWhcuM20W0wKMCp74m8vXFMnIOgUP3GsPoOkRcAokAZbF7K288nLXR6kaU7a00Qm7VPxtz5qATPiOsO76lFLSO5%2BqfbSGI0Vnsu%2FsyPRTZ31xplininVoEtWZgLTRlvVvoK%2FlOXMOSVsoAGOusBdx2c%2F9oHzJYrKKKXQ8uFXI%2FZ%2FSTgrQJMtwW%2FhL9nCFhsqhFnVmRAs2BwywAiUx%2BCqdjJWZWmTkRbCFT%2BHHSoNxjacBDPuZ6Bnx9Pc1%2Fm%2FmkLEP9neiaMKXPWFiLtfeWG8QApGokJM1eIB7WiyX7RSBz%2FeSbON1nmgU7rqgpoxAWWoFt5R1BtmSdLiaTRl%2Bkr%2F%2BcVJS2sZQ85FgvsdRTnrRMIJpKPdwCsVlhm7D7AA797ASt8bz2QkC8MsX149RprtoELP8iBrTNhcQALdmhQIO4tnWCCFYFLzmIlm7vlRwJ9Jc99bLr09S9vGA%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20210123T212144Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYQ2CR5ZGR%2F20210123%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=4497c8a33679d2b5cf46402ede6678840cc1506780ffff79f2854294ae02e1cd&hash=5ba89707ce0dcb87f4676f0f3411344167be8eebdc09980f6e9314345d8dc846&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=0021869380902331&tid=spdf-95359de1-2f03-4d31-abc4-e4e841532060&sid=7a58ddde1d338143876ab9766bc68efd0ffdgxrqb&type=client