How to prove that $\mathbb Z _p$ is compact?
where $\mathbb Z_ p = \{x \in \mathbb Q _p :\| x \|_p \leq 1\}$ is the set of p-adic integers.
Thanks in advance.
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2There are hints in the book #A Course in the Arithmetic# of J.P Serre. $\mathbb Z_p$ inherits a topology of $\prod \mathbb Z/p^n\mathbb Z$ if we define the discrete topology of $\prod \mathbb Z/p^n\mathbb Z$. We just need to prove that $\prod \mathbb Z/p^n\mathbb Z$ is compact and $\mathbb Z_p$ is closed. – gaoxinge Jan 20 '14 at 06:21
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You can find it here from the link: people.reed.edu/~jerry/361/lectures/zpcompact.pdf – DeepSea Jan 20 '14 at 06:51
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See this question: Why are closed balls in the $p$-adic topology compact?
Note, that everywhere in the answer $\frac{x_k}{p^k}$ must be changed to $x_k p^k$
