Is it possible that $L$ is not totally ramified over $K$?
Solution 1:
(I assume that all occuring fields are local, and that all extensions are finite.)
We have $$f_{L \mid \mathbb{Q}_p} = f_{L \mid K} \cdot f_{K \mid \mathbb{Q}_p},$$ and $L \mid \mathbb{Q}_p$ being totally ramified means by definition that $f_{L \mid \mathbb{Q}_p} = 1$. Since the $f$'s are positive integers, this is possible only if $f_{L \mid K} =1$, thus if $L$ is totally ramified over $K$.