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. $e=\sum_{n\geq 0}1/n!\notin\overline{\mathbb{Q}}$ because e is transcendental). The completion of $\overline{\mathbb{Q}}$ with respect to $|\cdot|$ is $\mathbb{C}$, the field of complex numbers.

## Construction of $\mathbb{C}_{p}$

We follow the construction of $\mathbb{C}$ above to build $\mathbb{C}_{p}$. 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 $p\equiv 1\mod 4$). 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 $\overline{\mathbb{Q}}_{p}$ is not complete with respect to $|\cdot|_{p}$.

###### Proof.

Let $\beta_{n}$ be defined as:

 $\beta_{n}=\begin{cases}e^{2\pi i/n},\text{ if }(n,p)=1;\\ 1,\text{ otherwise.}\end{cases}$

One can prove that if we define:

 $\alpha=\sum_{n=1}^{\infty}\beta_{n}p^{n}$

then $\alpha\notin\overline{\mathbb{Q}}_{p}$, although $\sum_{n=m}^{\infty}\beta_{n}p^{n}\to 0$ as $m\to\infty$ (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 $\overline{\mathbb{Q}}_{p}$ with respect to the $p$-adic absolute value $|\cdot|_{p}$.

###### Proposition (Properties of $\mathbb{C}_{p}$).

The field $\mathbb{C}_{p}$ enjoys the following properties:

1. 1.

$\mathbb{C}_{p}$ is algebraically closed.

2. 2.

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

3. 3.

$\mathbb{C}_{p}$

4. 4.

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

5. 5.

$\mathbb{C}_{p}$ is isomorphic to $\mathbb{C}$ as fields, although they are not isomorphic as topological spaces.

## References

Title complex p-adic numbers ComplexPadicNumbers 2013-03-22 15:13:44 2013-03-22 15:13:44 alozano (2414) alozano (2414) 6 alozano (2414) Definition msc 12J12 msc 11S99 complex $p$-adic numbers