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 might be related to the set of exponents of $p$ in a number's $p$-adic expansion, and in particular if is a discrete subset (or additively generates a discrete subgroup) of ${\bf Q}$ or not. I don't have much familiarity with the field structure of extensions of ${\bf Q}_p$ however (this discussion occurred in what is an introductory class in the $p$-adic numbers following Gouvea, in fact). Is the picture easier to describe if we use Teichmüller representatives?

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).

There might be 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, though; it's hard to imagine there being conditions on the decimal expansion of a real number that determines if it is 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 ${\bf Q}$ generated 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 have approximately zero familiarity with the field structure of extensions of ${\bf Q}_p$ 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. Also, perhaps highlighting the differences would be easier if we use Teichmüller representatives?