I've only ever seen finite field extensions indicated as $[L:K] < \infty$. I've never seen $-\infty<[L:K]<\infty$. I take this to mean that field extensions of a negative degrees are not considered.
This makes sense, because I have no idea how you would get a field extension of a negative degree. However, I'm curious, is there an interpretation of field theory that would allow for this? If so, what would a negative degree mean?