Fix a prime number $p$. Suppose that I have a valuation $v\_p: \mathbb{Q} \to \mathbb{Q}$ on the rationals $\mathbb{Q}$. That is, $v\_p( p^n(\frac{a}{b})) = p^{-n}$ where each of $a,b$ is not divisible by $p$. 


How can I extend $v\_p$ to $v$ on the reals $\mathbb{R}$ such that $v|\_\mathbb{Q} = v\_p$? I am looking for an explicit description of $v$, if that is possible. I know for a fact that one can extend valuation on any field extension.


Thank you, 



Extension of valuation
As you point out, it follows on general grounds that there is an extension of $v\_p$ to a valuation on ${\mathbb R}$ (in fact, there are uncountably many such extensions), but it is impossible to give an "explicit" description. Indeed, not only will any such extension by discontinuous with respect to the usual Euclidean topology on the reals, but it will not be a measurable function. 


