Finite separable simple extension implications
= Finite separable simple extension implications
A finite separable extension is simple by the primitive element theorem. A simple separable extension is finite because its generator is separable and hence algebraic. Finite and simple does not imply separable: in characteristic $p$, $\mathbb F_p(t^{1/p})/\mathbb F_p(t)$ is finite and simple but purely inseparable.