= Solution
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 $[L:K]=\deg m_{\alpha,K}<\infty$.
Finite and simple do not imply separable. In characteristic $p$, take
$$
K=\mathbb F_p(t),\qquad L=K(t^{1/p}).
$$
Then $L/K$ has degree $p$ and is simple, but the generator has inseparable minimal <polynomial> $X^p-t$. These are the <finite separable simple extension implications>.
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