Let $p$ be a prime such that $p>5$, $e\in \{1,2,3,4,6\}$, and $K_e$ be a totally ramified extension of $\mathbb{Q}_p$ of ramification index $e$.
If $K_{e'}$ is another Galois extension of $\mathbb{Q}_p$ of ramification index $e$, then how to prove there exists a finite unramified extension $M$ of $\mathbb{Q}_p$ such that $MK_e=MK_e'$?
More general, if we only assume $K_e$ is an finite extension of $\mathbb{Q}_p$ of ramification index $e$(not a totally ramified extension), do we have the same conclusion?
Remark: We fix an algebraic closure of $\mathbb{Q}_p$, and we do all of these in it.
Thanks!