Value group of the $p$-adic complex numbers

Question

I have read that the value group of the $p$-adic complex numbers ($\mathbb{C}_p$) and the algebraic closure of $\mathbb{Q}_p$ is $\mathbb{Q}$. I understand why the value group of $\mathbb{Q}_p$ is $\mathbb{Z}$, but fail to see how to relate this to the other two fields. More generally what is the relationship between the value group of a Non-Archimedean (NA) field and its algebraic closure.

Answer 1

To elaborate on stillconfused's comment, note that for any finite extension $K/\mathbb Q_p$, the value group is $\frac1{e(K)}\mathbb Z$, where $e(K)$ is the ramification degree. Thus, since the algebraic closure $\overline{\mathbb Q}_p$ is the union $\bigcup_{K/\mathbb Q_p}K$ where $K/\mathbb Q_p$ runs over all the finite extensions, the value group for $\overline{\mathbb Q}_p$ is $\bigcup_{K/\mathbb Q_p}\frac1{e(K)}\mathbb Z$. Thus, it suffices to check for each integer $n>0$ there is some field $K/\mathbb Q_p$ with $e(K)=n$. Note that $K=\mathbb Q_p(p^{1/n})$ is such an example.

Now, since $\mathbb C_p$ is the $p$-adic completion of $\overline{\mathbb{Q}}_p$, its character group is also $\mathbb Q$.