Product of two regular varieties over an imperfect field I am trying to find a counterexample to the following, but am unable to find one. Any help would be appreciated, and also an explanation of why it works. I am trying to show that over an imperfect field k, the fiber product of two regular varieties can fail to be regular. For an algebraically closed field, every product of regular varieties is regular, but the same shouldn't hold for imperfect.

As $k$ is imperfect, say of characteristic $p$, there exists $\alpha\in k$ which is not a $p$-th power in $k$. Consider $K=k[X]/(X^p-\alpha)$. This is a field, finite over $k$, so it defines a regular variety $V$.

Now consider the fiber product $V\times_k V$. Its ring of regular functions is $$K\otimes_k K=K\otimes_k k[X]/(X^p-\alpha)=K[X]/(X^p-\alpha)=K[X]/(X-\theta)^p $$ where $\theta\in K$ is the class of $X$ in $K$. This shows that $V\times_k V$ not regular, it is not even reduced !