PlanetMath: complex p-adic numbers

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complex p-adic numbers

(Definition)

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 $\vert\cdot\vert$ , e.g. $\vert-3/2\vert=3/2$ . 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 $\vert\cdot\vert$ (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 $\vert\cdot \vert$ 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 be a prime number and let $\mathbb{Q}_p$ be the -adic rationals or (-adic numbers). The -adics, $\mathbb{Q}_p$ , are the completion of $\mathbb{Q}$ with respect to the usual -adic valuation $\vert\cdot\vert _p$ . Thus, we regard $(\mathbb{Q}_p, \vert\cdot\vert _p)$ 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 -adic valuation $\vert\cdot\vert _p$ extends uniquely to $\overline{\mathbb{Q}}_p$ . However:

Proposition 1 The field $\overline{\mathbb{Q}}_p$ is not complete with respect to $\vert\cdot\vert _p$ .

Proof. Let $\beta_n$ be defined as:

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

One can prove that if we define:

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

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 $\vert\cdot\vert _p$ . $\qedsymbol$

Definition 1 The field of complex -adic numbers is defined to be the completion of $\overline{\mathbb{Q}}_p$ with respect to the -adic absolute value $\vert\cdot\vert _p$ .

Proposition 2 (Properties of $\mathbb{C}_p$ ) The field $\mathbb{C}_p$ enjoys the following properties:

$\mathbb{C}_p$ is algebraically closed.
The absolute value $\vert\cdot\vert _p$ extends uniquely to $\mathbb{C}_p$ , which becomes an algebraically closed, complete metric space.
$\mathbb{C}_p$ is a complete ultrametric field.
$\overline{\mathbb{Q}}_p$ is dense in $\mathbb{C}_p$ .
$\mathbb{C}_p$ is isomorphic to $\mathbb{C}$ as fields, although they are not isomorphic as topological spaces.

Bibliography

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

"complex p-adic numbers" is owned by alozano.

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Other names:

complex

-adic numbers

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Attachments:: p-adic exponential and p-adic logarithm (Definition) by alozano
p-adic analytic (Definition) by alozano

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Cross-references: topological spaces, isomorphic, dense in, complete ultrametric field, properties, complex, valuation, numbers, prime number, completion, e is transcendental, complete, algebraic closure, fixed, algebraically closed, field, absolute value, distance, metric space, rational numbers, complex numbers
There are 2 references to this entry.

This is version 3 of complex p-adic numbers, born on 2005-05-02, modified 2005-05-02.
Object id is 6998, canonical name is ComplexPAdicNumbers5.
Accessed 1766 times total.

Classification:

AMS MSC:	11S99 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Miscellaneous)
	12J12 (Field theory and polynomials :: Topological fields :: Formally $p$-adic fields)

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