In his paper "The concept of a simple point of an abstract algebraic variety" Oscar Zariski provides in Example 1 a regular but not smooth variety over a field $K$.
In his example $K$ is a non-perfect field of characteristic $p$ and $$X = \operatorname{Spec} K[X,Y]/(X^p + Y^p - a),$$ where $a$ is an element of $K$ which is not a $p$-th power.
I see that $X$ is not smooth since the Jacobian matrix is $(0)$ and therefore not of maximal rank. I also know that the example would not work if $K$ was perfect.
But I have problems proving the regularity of $X$. Zariski mentions that the curve is (obviously?) normal, but I don't see why and in particular I don't know where the non-perfectness of $K$ has to be used.
Thanks in advance.
After making the change of variables $(X,Y) \mapsto (X,X+Y)$, we have that $X$ is isomorphic to $K[X,Y]/(X^{p}-a)$. Setting $L := K[X]/(X^{p}-a)$, which is a purely inseparable field extension of $K$, we have $K[X,Y]/(X^{p}-a) \simeq L[Y]$, which is a regular ring but is not smooth over $K$.