2 corrected typo

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 $v|_\mathbb{R} v|_\mathbb{Q} = v_p$? 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,

1

# Extension of valuation

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 $v|_\mathbb{R} = v_p$? 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,