This is a question is from a past paper, and it's only worth 8 marks, and it's really got me. I'm allowed to assume that if $L/K$ is such an extension, that there are $[L:K]$ $K$-embeddings of $L$ into $\mathbb{C}$.
I was thinking the only way to do this would be to show some $\alpha$ such that $L=K(\alpha)$ exists, though I have no ideas about how to. The hint doesn't seem to help me much; the $K$-embeddings of a simple extension are completely determined by what they do to the adjoined element, but how could I use this? Knowing how many embeddings there are doesn't seem to help show that such an element exists.
By the primitive element theorem, finite separable extensions are simple. This doesn't use the hint, though.