Let $p\in{X}$ where $X$ is a curve-- here the definition of a curve is an integral, seperated, 1 dimensional scheme of finite type over a field $k$ (not necessarily algebraically closed). Moreover, suppose $X$ is smooth at $p$, so that the sheaf of differentials $\Omega_{X/k}$ is free of rank 1 on some neighborhood of $p$. From this alone, can we conclude that $X$ is regular at $p$, i.e. is $\mathcal{O}_{X,p}$ is a regular local ring?
I realize that the answer is yes if $k$ is perfect, or if $k(p)$, the function field at p, is just $k$, or even if $\mathcal{O}_{X,p}$ contains a subfield isomorphic to $k(p)$, but I can find absolutely nothing in the literature without these restrictions. Can anyone set me straight here?