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. 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