Why is the trace map from a finite extension of $\mathbb{Q}_p$ continuous?

Question

Let $k$ be a finite extension of $\mathbb{Q}_{p}$. Why is $tr_{k/\mathbb{Q}_{p}}$ a continuous map from $k$ onto $\mathbb{Q}_{p}$?

Answer 1

Show that a $\mathbb Q_p$-linear map $f:V\to W$ of finite dimensional $\mathbb Q_p$-vector spaces is always continuous. What you want then follows from $\mathbb Q_p$-linearity of the trace, which is more or less immediate.