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Let $K$ be a finite extension of $\mathbb Q_p$. How to prove that if an element of $K$ has non negative valuation then it is algebraic over $\mathbb Z_p$?

I would like also a reference for this proof (e.g. is it somewhere in Algebraic Number Theory by Neukirch ?).

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Rather than algebraic over $\mathbb Z_p$, probably integral is what is intended, in any case. – paul garrett Sep 15 '11 at 21:52
up vote 4 down vote accepted

Since you're talking about a valuation on $K$, I guess that you're aware of the fact that the absolute value on $\mathbf{Q}_p$ extends uniquely to $K$. The formula for the extended absolute value of $\alpha\in K$ is $\vert N_{K/\mathbf{Q}_p}(\alpha)\vert^{1/[K:\mathbf{Q}_p]}$. Using Hensel's Lemma it can be shown that the integral closure of $\mathbf{Z}_p$ in $K$ is precisely the set of $\alpha\in K$ with $N_{K/\mathbf{Q}_p}(\alpha)\in\mathbf{Z}_p$. This is explained in the beginning of the proof of Theorem 4.8 of Chapter II in Neukirch, and actually this fact logically precedes the fact that the valuation on $\mathbf{Q}_p$ extends uniquely to $K$ (this is actually the statement of Theorem 4.8). At any rate, once you have this description of the integral closure and the formula for the extended valuation, it follows immediately that the valuation ring of $K$, i.e., the set of elements with non-negative valuation, is the integral closure of $\mathbf{Z}_p$ in $K$. Neukirch even makes this observation at the end of the proof of the aforementioned theorem.

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