I'd like to fine an example of field $K$ and elements $\alpha, \beta$ such that $\mathrm{char}(K) = p> 0 $, $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$.
This obviously can't work if $K$ is a finite field. So I need to find a non-finite $K$. The only ones that pop into my head are $\mathbb F_p(t)$, $\mathbb F_p(t^p)$ and $\overline{\mathbb F}_p$ for $t$ an indererminate, but I'm struggling to find an example.
Any hints would be greatly appreciated.
Thanks