complex padic numbers
First, we review a possible construction of the complex numbers^{}. We start from the rational numbers, $\mathbb{Q}$, which we consider as a metric space, where the distance is given by the usual absolute value^{} $\cdot $, e.g. $3/2=3/2$. As we know, the field of rational numbers is not an algebraically closed field (e.g. $i=\sqrt{1}\notin \mathbb{Q}$). Let $\overline{\mathbb{Q}}$ be a fixed algebraic closure^{} of $\mathbb{Q}$. The absolute value in $\mathbb{Q}$ extends uniquely to $\overline{\mathbb{Q}}$. However, $\overline{\mathbb{Q}}$ is not complete^{} with respect to $\cdot $ (e.g. $e={\sum}_{n\ge 0}1/n!\notin \overline{\mathbb{Q}}$ because e is transcendental). The completion of $\overline{\mathbb{Q}}$ with respect to $\cdot $ is $\u2102$, the field of complex numbers.
Construction of ${\u2102}_{p}$
We follow the construction of $\u2102$ above to build ${\u2102}_{p}$. Let $p$ be a prime number^{} and let ${\mathbb{Q}}_{p}$ be the $p$adic rationals (http://planetmath.org/PAdicIntegers) or ($p$adic numbers). The $p$adics, ${\mathbb{Q}}_{p}$, are the completion of $\mathbb{Q}$ with respect to the usual $p$adic valuation (http://planetmath.org/PAdicValuation) $\cdot {}_{p}$. Thus, we regard $({\mathbb{Q}}_{p},\cdot {}_{p})$ as a complete metric space. However, the field ${\mathbb{Q}}_{p}$ is not algebraically closed (e.g. $i=\sqrt{1}\in {\mathbb{Q}}_{p}$ if and only if $p\equiv 1mod4$). Let ${\overline{\mathbb{Q}}}_{p}$ be a fixed algebraic closure of ${\mathbb{Q}}_{p}$. The $p$adic valuation $\cdot {}_{p}$ extends uniquely to ${\overline{\mathbb{Q}}}_{p}$. However:
Proposition.
The field ${\overline{\mathrm{Q}}}_{p}$ is not complete with respect to $\mathrm{}\mathrm{\cdot}{\mathrm{}}_{p}$.
Proof.
Let ${\beta}_{n}$ be defined as:
$${\beta}_{n}=\{\begin{array}{cc}{e}^{2\pi i/n},\text{if}(n,p)=1;\hfill & \\ 1,\text{otherwise.}\hfill & \end{array}$$ 
One can prove that if we define:
$$\alpha =\sum _{n=1}^{\mathrm{\infty}}{\beta}_{n}{p}^{n}$$ 
then $\alpha \notin {\overline{\mathbb{Q}}}_{p}$, although ${\sum}_{n=m}^{\mathrm{\infty}}{\beta}_{n}{p}^{n}\to 0$ as $m\to \mathrm{\infty}$ (see [1], p. 48, for details). Thus, ${\overline{\mathbb{Q}}}_{p}$ is not complete with respect to $\cdot {}_{p}$. ∎
Definition.
The field of complex $p$adic numbers is defined to be the completion of ${\overline{\mathrm{Q}}}_{p}$ with respect to the $p$adic absolute value $\mathrm{}\mathrm{\cdot}{\mathrm{}}_{p}$.
Proposition (Properties of ${\mathrm{C}}_{p}$).
The field ${\mathrm{C}}_{p}$ enjoys the following properties:

1.
${\u2102}_{p}$ is algebraically closed.

2.
The absolute value $\cdot {}_{p}$ extends uniquely to ${\u2102}_{p}$, which becomes an algebraically closed, complete metric space.

3.
${\u2102}_{p}$ is a complete ultrametric field.

4.
${\overline{\mathbb{Q}}}_{p}$ is dense in ${\u2102}_{p}$.

5.
${\u2102}_{p}$ is isomorphic to $\u2102$ as fields, although they are not isomorphic as topological spaces^{}.
References
 1 L. C. Washington, Introduction to Cyclotomic Fields^{}, SpringerVerlag, New York.
Title  complex padic numbers 

Canonical name  ComplexPadicNumbers 
Date of creation  20130322 15:13:44 
Last modified on  20130322 15:13:44 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  6 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 12J12 
Classification  msc 11S99 
Synonym  complex $p$adic numbers 