The answer to this 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) 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.