Is there a non-archimedian valuation on $\mathbb{C}$?

Question

I was wondering whether there can be a (non-trivial) non-archimedian valuation on $\mathbb{C}$ (with value group contained in $\mathbb{R}$). My idea of constructing such a valuation would be to take some transcendent element $t\in\mathbb{C}$. Then the discrete valuation ring $\mathbb{Q}[t]_{(t)}$ gives a valuation on $\mathbb{Q}(t)$. But I was not able to decide whether one can extend it to all of $\mathbb{C}$.

Answer 1

You can actually extend the $p$-adic valuation of $\Bbb Q$ to $\Bbb C$. Or you can use an isomorphism $\Bbb C\cong \overline{\Bbb Q_p}$.