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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy