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Let $K$ be a number field, $m=[K: Q]$ and the place $v$ is some extension of p-adic norm on field $Q$ of rational numbers for some prime number $p$, then for any $x\in K$, do we have
$$|x|_v<1~ ==> ~ |x|_v\leq \frac{1}{p^{1/m}}??$$

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Yes. If $v$ is some extension of $v_p(\cdot)$ then the maximal ideal of the valuation subring of $K$ is a prime ideal of ${\frak O}_K$ lying above $p$ with ramification index $e\le m$. The value group is $\frac{1}{e}\Bbb Z$. –  anon Oct 21 '13 at 5:19
Great. Thanks a lot! –  jocy Oct 22 '13 at 2:02

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