The answer to [this](https://mathoverflow.net/questions/371948/separability-of-mathbbcx-over-its-mathbbc-subalgebras/382150?noredirect=1#comment972149_382150) MO question says the following:


**Lemma 1.** *Let $h \in \mathbf C[x]$ be a polynomial of degree $n \geq 2$. Then $\mathbf C+(h) \subseteq \mathbf C[x]$ is unramified if and only if $h$ is squarefree.*


I wonder what happens in higher dimensions, namely:



> 
> **Question:** Is the following claim true: Let $h \in \mathbb{C}[x]$ be a polynomial of degree $n \geq 2$.
> Then $\mathbb{C}[x,y\_1,\ldots,y\_r]$ is unramified (equivalently: separable) over $\mathbb{C}+(h,y\_1,\ldots,y\_r)$ if and only if $h$ is squarefree.
> 
> 
> 


What I think is that if Lemma 1 can be generalized to the following Lemma 2, then the answer to my current question is positive.


**Lemma 2.** *Let $A$ be a commutative $\mathbb{C}$-algebra of characteristic zero.
Let $h \in A[x]$ be a polynomial of degree $n \geq 2$. Then $A +(h) \subseteq A[x]$ is unramified if and only if $h$ is squarefree.*


**Indeed:** If Lemma 2 is true, then $\mathbb{C}[y\_1,\ldots,y\_r]+(h) \subset \mathbb{C}[y\_1,\ldots,y\_r][x]= \mathbb{C}[x,y\_1,\ldots,y\_r]$ is unramified if and only if $h$ is aquarefree.


Recall a known result about separability (can be found [here](https://www.springer.com/gp/book/9783540053712)) which says that given $R\_1 \subseteq R\_2 \subseteq R\_3$, if $R\_1 \subseteq R\_3$ is separable, then $R\_2 \subseteq R\_3$ is separable.


Therefore here, if $h$ is squarefree, then $\mathbb{C}[y\_1,\ldots,y\_r]+(h) \subset \mathbb{C}[x,y\_1,\ldots,y\_r]$ is unramified, and then $\mathbb{C}+ (h,y\_1,\ldots,y\_r) \subset \mathbb{C}[x,y\_1,\ldots,y\_r]$ is unramified.


Any hints and comments are welcome! Thank you.



Separability of $\mathbb{C}[x,y_1,\ldots,y_r]$ over $\mathbb{C} + (h,y_1,\ldots,y_r)$
In Lemma 2, I'm not sure what "squarefree" means. However, the meaning is clear if $h\in\mathbb{C}[x]$, and then Lemma 2 is true. Indeed, the inclusion $\mathbb{C}+(h)\subset \mathbb{C}[x]$ induces an injection $A\otimes\_\mathbb{C}(\mathbb{C}+(h))\subset A[x]$. The image is easily checked to be $A+(h)$. The lemma follows because being unramified is preserved by base change.


