"Purely inseparable, algebraic extension if and only if trivial automorphism group"

Question

Let $k \subset K$ be an algebraic field extension, with $\mathrm{char} \, k=p>0$. Is it true the following assertion?

$k \subset K$ is purely inseparable if and only if $\mathrm{Aut}_k(K)=\{1\}$; where $\mathrm{Aut}_k(K)$ are the automorphisms of $K$ which fix $k$. I showed the arrow $\Rightarrow$ and i think it works. What about the arrow $\Leftarrow$?.

Thanks to everyone.

Answer 1

If you take $K=\Bbb F_5(t)$ and $k=\Bbb F_5(t^3)$ then the extension is separable but there are no automorphisms as there are no 3rd roots of $1$ in $k$ (besides $1$).