complex p-adic 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. $e=\sum_{n\geq 0}1/n!\notin\overline{\mathbb{Q}}$ 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.

$\mathbb{C}_{p}$ is algebraically closed.
2.

The absolute value $|\cdot|_{p}$ extends uniquely to $\mathbb{C}_{p}$ , which becomes an algebraically closed, complete metric space.
3.

$\mathbb{C}_{p}$ is a complete ultrametric field.
4.

$\overline{\mathbb{Q}}_{p}$ is dense in $\mathbb{C}_{p}$ .
5.

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

References

1 L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.

Title	complex p-adic numbers
Canonical name	ComplexPadicNumbers
Date of creation	2013-03-22 15:13:44
Last modified on	2013-03-22 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