Let $K$ be a imperfect field. Let $L/K$ be a finite field extension.
Is $L$ imperfect?
Suppose that $L/K$ is separable. Is $L$ imperfect?
Suppose that $L/K$ is Galois. Is $L$ imperfect?
I'm looking for examples, counterexamples, explanations, and proofs of course.
Note: The converse holds. If $K$ is perfect, then any finite field extension of $K$ is perfect.