I am trying to prove a statement about the smoothness of a scheme (it is indeed the exercise 6.2.2 in Qing Liu's "Algebraic geometry and Arithmetic Curve"):
Let $X$ be a (Noetherian) scheme of locally of finite type over a field $k$. If $\Omega_{X/k}$ is locally free, and if $k(\xi)$ is a separable extension of $k$ for every generic point $\xi$, then $X$ is smooth.
(Separable extension means $k(\xi)$ is a finite extension of a purely inseparable extension of $k$).
Here is my attempt: If $X$ is reduced at $\xi$, then $\mathcal{O}_{x, \xi}=k(\xi)$ and $\Omega_{X, \xi}=\Omega_{k(\xi)/k}$. Since $k(\xi)/k$ is separable, $\Omega_{k(\xi)/k}$ is a finite dimensional $k(\xi)$-vector space and $$\dim_{k(\xi)}\Omega_{X, \xi}=\text{trdeg}_{k}(k(\xi))=\dim_{\xi}X,$$ so $X$ is smooth at $\xi$. By "spreading out" we can show the smoothness at every point of $X$. However, I am stuck in trying to prove that $X$ must be reduced at $\xi$. I guess that it is related to the property of "separable extension", but I don't know how to use it.
Can anybody help me? Thank you so much.