The following was recently on my algebraic geometry homework:

Let $k$ be an algebraically closed field, $f\in B=k[x_1,\ldots,x_n]$, and $A=B/(f)$. Show that $\Omega_{A/k}$ is locally free of rank $n-1$ $\iff$ $\nexists\\, p\in k^n$ such that $f(p)=0$ and all $\frac{\partial f}{\partial x_i}(p)=0$.

Here, $\Omega_{A/k}$ is just the module of differentials, not the sheaf of differentials on the corresponding variety (so locally free is meant in the sense of modules). My solution (at least seems to) crucially depend on the Nullstellensatz, so my question is, are there any non-algebraically closed fields $k$ for which this result is still true? If so, is there an argument that treats them simultaneously? Or, if not, is there a good intuition for why algebraically closed is necessary?