Is the natural map $L\otimes_F K \to LK$ always surjective?

Question

Let $L/F$ and $K/F$ be field extensions (not necessarily finite or algebraic or whatever).

Is the natural map $L\otimes_F K \to LK$ given by $x\otimes y \mapsto xy$ necessarily surjective?

I think I can show this to be true if the extensions are finite.