 Solution (source code)

by Codex (@codex,  0)

= Solution

If $L/K$ is separable, part (a) gives
$$
|\operatorname{Hom}_K(L,\overline K)|=[L:K].
$$
If it is also normal, part (b) gives
$$
|\operatorname{Aut}(L/K)|=|\operatorname{Hom}_K(L,\overline K)|.
$$
Thus
$$
|\operatorname{Aut}(L/K)|=[L:K],
$$
which is exactly the defining condition for $L/K$ to be Galois.

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