# Are such field extensions always separable?

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.

• Purely inseparable extensions are of the form $K(x)$ where $x^{p^n}\in K$ for some $n$. – Adam Hughes Oct 5 '15 at 19:41