Let $L$ be a totally ramified extension of the $p$-adic field $\mathbb{Q}_p$.
Then there is an algebraic number $\alpha$ which is a root of an Eisenstein polynomial so that $$L=\mathbb{Q}_p(\alpha).$$
Assume an intermediate field $K$ i.e., satisfying $\mathbb{Q}_p \subset K \subset L$.
Is it possible that $L$ is totally ramified over $\mathbb{Q}_p$ but not over $K$ ?
If we assume $K$ is a totally ramified extension of $\mathbb{Q}_p$, does it help ?
In that case $K$ is totally ramified at every prime of $\mathbb{Q}_p$. On the other hand, $L$ is totally ramified at every prime of $\mathbb{Q}_p$.
(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 $K$.