My question is quite simple: Let $X$ be an irreducible algebraic variety over a field $\Bbbk$. Is there a name for such varieties with perfect function field $\Bbbk(X)$? Is this very rare? Is there literature that deals with this kind of question and object?



Variety with perfect function field?
This is a pretty easy question, but I don't see why not to record an answer. (The answer was clearly known to user ACL...as it would be to almost any arithmetic geometer of his pay grade.) 


Case 1: $k$ characteristic $0$. Then $k(V)$ is always perfect, of course.


Case 2: $k$ has characteristic $p > 0$ but is perfect. Then $k(V)$ is perfect iff $\operatorname{dim} V = 0$ (iff $V = \operatorname{Spec} k$ for a finite field extension of $l/k$). Let us use the criterion that if $L/K$ is a finite degree field extension in positive characteristic, then $K$ is perfect if and only if $L$ is perfect (I'll leave this as an exercise for now). This immediately gives one implication. For the other, we write $L = k(V)$ as a finite extension of a purely transcendental field $K = k(t\_1,\ldots,t\_n)$ (you can view this either as pure field theory or as a weak version of Noether normalization) and observe that if $n \geq 1$ then the indeterminates $t\_i$ are certainly not $p$th powers in $K$. 


Case 3: $k$ is not perfect. Then $k(V)$ is never perfect. The argument of Case 2 carries over immediately.


Please do feel free to ask for more details if necessary. 


