Let $k$ be a number field, $v$ a discrete non-archimedean valuation on $k$. Let $k_v$ be the completion of $k$ wrt $v$ and $\mathcal O_v$ its valuation ring.
My question is:
If $x\in k_v$ then is there an integer $m\in \mathbb{Z}$ (not merely $x\in \mathbb{Z}_p$) such that $mx\in \mathcal{O}_v?$
Many thanks in advance!
Hint: $v$ corresponds to a prime ${\frak p}$ of $k$. Then ${\frak p}$ lies above $(p)$ for some rational prime $p$. Thus we have an integer with positive valuation...