I am trying to construct a finite field extension $L/K$ of prime characteristic $p$, such that $\gcd([L:K],p)=1$, and that has an inseparable element.
Trying to construct an example of such extension, I could only find infinite extensions and extensions where $\gcd([L:K],p)\neq1$, but I can neither prove that the mentioned conditions define a separable extension.
please help :)