First, we review a possible construction of the complex numbers. We start from the rational numbers, $\Rats$ , 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. $i=\sqrt{-1} \notin \Rats$ ). Let $\overline{\Rats}$ be a fixed algebraic closure of $\Rats$ . The absolute value in $\Rats$ extends uniquely to $\overline{\Rats}$ . However, $\overline{\Rats}$ is not complete with respect to $|\cdot|$ (e.g. $e=\sum_{n\geq 0} 1/n!\notin \overline{\Rats}$ because e is transcendental). The completion of $\overline{\Rats}$ with respect to $|\cdot |$ is $\Complex$ , the field of complex numbers.

Construction of $\Complex_p$

We follow the construction of $\Complex$ above to build $\Complex_p$ . Let $p$ be a prime number and let $\Rats_p$ be the $p$ -adic rationals or ($p$ -adic numbers). The $p$ -adics, $\Rats_p$ , are the completion of $\Rats$ with respect to the usual $p$ -adic valuation $|\cdot|_p$ . Thus, we regard $(\Rats_p, |\cdot|_p)$ as a complete metric space. However, the field $\Rats_p$ is not algebraically closed (e.g. $i=\sqrt{-1}\in \Rats_p$ if and only if $p \equiv 1 \mod 4$ ). Let $\overline{\Rats}_p$ be a fixed algebraic closure of $\Rats_p$ . The $p$ -adic valuation $|\cdot|_p$ extends uniquely to $\overline{\Rats}_p$ . However:

Proof. Let $\beta_n$ be defined as:

One can prove that if we define: $$\alpha=\sum_{n=1}^\infty \beta_n p^n$$ then $\alpha\notin \overline{\Rats}_p$ , although $\sum_{n=m}^\infty \beta_n p^n \to 0$ as $m\to \infty$ (see [1], p. 48, for details). Thus, $\overline{\Rats}_p$ is not complete with respect to $|\cdot|_p$ .

Bibliography

1

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