I was discussing the tree picture of ${\bf Z}_p$ and ${\bf Q}_p$ and mentioned that the idea can be extended to ${\bf C}_p$, with the caveat that the tree is no longer locally finite (as the value group $\bf Q$ is not discrete like ${\bf Z}$ is), and that there is no longer a root. My professor asked a question I hadn't considered: when we look at the extended picture with ${\bf C}_p$, is there a way to distinguish between algebraic and transcendental values simply by how they look as leaves of the tree? By extension we can also consider the difference between general elements of ${\bf C}_p$ and (say) the spherical completion $\Omega_p$ (see SBSeminar for a definition).
My intuition is that it
There might be related to necessary and sufficient conditions (for e.g. being in ${\bf C}_p\setminus \overline{{\bf Q}_p},\Omega_p\setminus{\bf C}_p$) on the set of exponents of $p$ in a number's $p$-adic expansion. This might be a stretch, and in particular though; it's hard to imagine there being conditions on the decimal expansion of a real number that determines if it is a discrete subset (or transcendental.
However, if $x=\sum_{\ell\ge u/v}a(\ell)p^\ell$ and the support of $a(\cdot)$ additively generates a discrete and hence cyclic subgroup ) of ${\bf Q}$ or notgenerated by say $r/s$ then we can write $x$ as a ${\bf Q}_p$-linear combination of the numbers $1,p^{1/s},p^{2/s},\cdots,p^{1-1/s}$ and hence $x\in{\bf Q}_p(p^{1/s})$ is algebraic. Is there a converse?
I don't have much approximately zero familiarity with the field structure of extensions of ${\bf Q}_p$ however unfortunately (this discussion occurred in what is an introductory class in the $p$-adic numbers following Gouvea, in fact), so I'd appreciate input from anyone with more background. Is Also, perhaps highlighting the picture differences would be easier to describe if we use Teichmüller representatives?
