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

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Have you tried something? –  Patrick Da Silva Mar 19 '12 at 3:00
    
You need to put together a few facts, but this shouldn't be so bad. What's your favorite definition of the trace? –  Dylan Moreland Mar 19 '12 at 3:03
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The key fact here is that the trace map is linear. –  Álvaro Lozano-Robledo Mar 19 '12 at 3:16

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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.

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