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Finite and separable imply simple by the primitive element theorem. Separable and simple imply finite: if $L=K(\alpha )$ and the extension is separable, then $\alpha$ is algebraic, so $[L:K]=\mathrm{deg}{m}_{\alpha ,K}<\infty$.

Finite and simple do not imply separable. In characteristic $p$, takeThen $L/K$ has degree $p$ and is simple, but the generator has inseparable minimal polynomial $X^p-t$. These are the finite separable simple extension implications.