The field $\mathbb{C}\_p$ of $p$-adic complex numbers is the completion of the algebraic closure of $\mathbb{Q}\_p$ with the corresponding extension of the usual non-Archimedean valuation $|\;\;|\_p$.


Every $x\in\mathbb{Q}\_p$ has a unique representation of the form $\sum\_{i=m}^\infty a\_ip^i$, where $m\in\mathbb{Z}$ and the $a\_i$'s are representatives of the classes in $\mathbb{Z}/p\mathbb{Z}$.
Does $y\in\mathbb{C}\_p$ have a similar representation as a generalized power series? Any reference? 



convergent series representation for p-adic complex numbers
The blog post <http://sbseminar.wordpress.com/2007/08/21/p-adic-fields-for-beginners> gives a good overview of the situation.


To briefly summarize (extracted from the above post — any errors are probably mine):


* The elements of $\mathbb{Q}\_p$ are exactly those represented by series of the form $\sum\_{r \in S} a\_r p^r$, where $S$ is a bounded-below subset of the integers and each $a\_r$ is a multiplicative lift (i.e., a "Teichmüller lift") of an element of $\mathbb{F}\_p$.
* The elements of $\mathbb{Q}\_p^{nr}$ (the maximal unramified extension of $\mathbb{Q}\_p$) are exactly represented by series of the form $\sum\_{r \in S} a\_r p^r$, where $S$ is again a bounded-below subset of the integers, but the $a\_r$ can now be lifts of any element of $\mathbb{F}\_q$, where $q$ is a fixed power of $p$. (As explained in the comments below, the $a\_r$ cannot be lifts of any elements of $\bar{\mathbb{F}}\_p$; they must all be contained in a common finite extension of $\mathbb{F}\_p$. This constraint is removed in the completion of $\mathbb{Q}\_p^{nr}$.)
* The elements of $\bar{\mathbb{Q}}\_p$ and its completion $\mathbb{C}\_p$ are represented by certain series of the form $\sum\_{r \in S} a\_r p^r$, where the $a\_r$ are again lifts of elements of $\bar{\mathbb{F}}\_p$, but now $S$ can be a more general well-ordered subset of $\mathbb{Q}$. Which such series give rise to elements of $\mathbb{C}\_p$ is discussed in [Kedlaya, *Power series and $p$-adic algebraic closures*](http://arxiv.org/abs/math/9906030) ([MR](http://www.ams.org/mathscinet-getitem?mr=1845241)).
* The elements of $\Omega\_p$, the *spherical completion* of $\mathbb{C}\_p$, are represented exactly by series of the form $\sum\_{r \in S} a\_r p^r$, where the $a\_r$ are as before, but now $S$ can be *any* well-ordered subset of $\mathbb{Q}$ (with no other restrictions). This is attributed to the undergraduate thesis of Bjorn Poonen, written up in [Poonen, *Maximally complete fields*](http://www-math.mit.edu/~poonen/papers/amsval.pdf) ([MR](http://www.ams.org/mathscinet-getitem?mr=1225257)).


And this is in some sense as far as we can go, since $\Omega\_p$ is "maximally complete": we can't extend it any further without "adding geometry", in the sense that it's the unique largest field of characteristic zero with value group $\mathbb{Q}$ and residue field the algebraic closure of $\mathbb{F}\_p$.


