Let $L$ and $L'$ be finite extensions of $K = \mathbb{Q}_p$. Also, let $n = [L:K]$ and $e = e(L/K)$. Furthermore, we assume the following properties:
- $L$ and $L'$ are both cyclic over $K$,
- $L'/K$ is totally ramified of degree $e$,
- $LL'/L'$ is unramified of degree $n$.
Question Is it true that $L \cap L' = K$?
I think this is true but I cannot prove why. I also tried to consider the case where $n$ is a prime power but it did not work out. All I know is that $L \cap L'/K$ as both $L$ and $L'$ contain $K$.
Could someone please help me to answer my question? Thank you again!