Let $K$ be a field with non-Archimedean valuation $|\cdot|$. Suppose that $\mathbb{R}\subset K$.
Question 1: Is the restriction of $|\cdot|$ to $\mathbb{R}$ the trivial valuation?
I guess that the answer is yes, but I don't see an evident answer. When I try to organize my ideas I get other questions:
Definition: Two valuations $|\cdot|_1$ and $|\cdot|_2$ are dependent if there exists $\lambda>0$ such that $|\cdot|_1=|\cdot|_2^{\lambda}$.
According Ostrowski's theorem, if the restriction of $|\cdot|$ to $\mathbb{Q}$ is not trivial, then it is dependent on the p-adic valuation.
Question 2: How to prove that $|\cdot|\neq|\cdot|_p$ on $\mathbb{Q}$?
Question 3: If $|\cdot|$ is trivial on $\mathbb{Q}$, how to prove that $|\cdot|$ is trivial on $\mathbb{R}$?
No, the restriction of $|\cdot|$ to $\mathbb{R}$ need not be trivial. For instance, the valuation on $\mathbb{Q}_p$ extends to a valuation on its algebraic closure $\overline{\mathbb{Q}_p}$. We can choose a field isomorphism $\mathbb{C}\to\overline{\mathbb{Q}_p}$ to then get a valuation on $\mathbb{C}$ whose restriction to $\mathbb{Q}$ is the $p$-adic valuation.