Let $K=\mathbb{Q}_p$ for some prime $p$. I would like to find an example for an extension $L/K$ with the following properties (or check if such an example is possible in the first place):
- $L/K$ is a cyclic extension,
- $L/K$ has degree $n=q^k$ where $q$ is also a prime and $k$ is some positive integer,
- $L/K$ is not unramified,
- $L/K$ is not totally ramified.
Here my thoughts:
- I know that $k=1$ won't work (as $L/K$ would be either unramified or totally ramified then), so $k \geq 2$.
- I am not sure whether I must assume $p=q$ in order for my example to work.
- And last, I know that $K$ has a primitive $n$-th root of unity (which I doubt we can always assume), Kummer Theory says that if $L/K$ is cyclic, then $L=K(\alpha)$ for some $\alpha\in L$ with $\alpha,\dots,\alpha^{n-1} \not\in K$ but $\alpha^n \in K$.
I am not sure if these thoughts are helpful for me to find an example at all. Could you please guide me through my problem?