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I'm taking a shot in the dark with this question, so I apologize if it makes no sense. Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $K_n$ be the field obtained by adjoining the $n$-th ...

Let $K$ be a finite extension of $Q_p$. Let $q$ be the size of the residue field of $K$, and let $\pi$ be a uniformizer of $K$. Then $q/\pi$ is some power of $\pi$ up to a unit $u$ in $K$, say $q/\pi =...