What is the value group of $\overline {\Bbb{Q}_p}$ and $ \Bbb{C}_p$ ? And are they discrete?
For finite extension of $ \Bbb{Q}_p$, there are known results for extension of valuations, but what about infinite case ? I know $ \Bbb{C}_p$'s value group is not discrete one, but I want to see what is exactly the value group, what is $｜ \Bbb{C}_p｜$ as a set.
Thank you in advance.
P.S.
Zerox gave me ｜$\Bbb{C}_p$｜＝{$p^a$｜$a∈\Bbb{Q}$}$∪${$0$}(Thank you so much). Could anyone give reference for this result or self-contained proof here ?
The value group $\Gamma$ of an algebraically closed valued field $(K,v)$ is always divisible: let $\gamma=v(x)\in \Gamma$; then for any $n\in \mathbb{N}$, $x$ has an $n$th root $y\in K$, so $\gamma=v(y^n)=nv(y)$ is $n$-divisible.
More precisely, the value group of the algebraic closure of some valued field is always the divisible hull of the value group. This should be done in any book about valuations, I'm pretty sure it is in the book by Efrat.
In the case of $\overline{\mathbb{Q}_p}$ and $\mathbb{C}_p$, the value group is $\mathbb{Q}$.