Let $k$ be a finite extension of $\mathbb{Q}_{p}$. Why is $tr_{k/\mathbb{Q}_{p}}$ a continuous map from $k$ onto $\mathbb{Q}_{p}$?
Why is the trace map from a finite extension of $\mathbb{Q}_p$ continuous?
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algebraic-number-theory
galois-theory
p-adic-number-theory
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1The key fact here is that the trace map is linear. – 2012-03-19
1 Answers
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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.