# Is $K[x_1,\ldots,x_{n+1}]$ separable over $K[x_1,\ldots,x_n]$?

Let $R \subseteq S$ be commutative rings. $S$ is separable over $R$ if $S$ is a projective $S \otimes_R S$-module (under $\mu: S \otimes_R S \to S$ defined by $\mu(s_1 \otimes s_2)=s_1s_2$).

Let $K$ be a zero characteristic field. Is $K[x_1,\ldots,x_{n+1}]$ separable over $K[x_1,\ldots,x_n]$ ?

• I think the answer is NO, due to a result of Wang, in his paper "Jacobian criterion for separability", Corollary 8, taking: $A=K[x_1,\ldots,x_n]$, $B=K[x_1,\ldots,x_n,x_{n+1}]$, $h(Z)=0$. – user237522 Jun 9 '15 at 0:40

Let $A=K[x_1,\ldots,x_n]$, and $B=A[Z]$ (to keep the notation from Wang, Corollary 8 of A Jacobian criterion for separability). Then $B=A[Z]/(h(Z))$ with $h=0$. If $B$ were separable over $A$, the above mentioned result of Wang entails $h'(Z)=0$ invertible in $B$, a contradiction.