complex p-adic numbers

First, we review a possible construction of the complex numbersMathworldPlanetmathPlanetmath. We start from the rational numbers, , which we consider as a metric space, where the distance is given by the usual absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath ||, e.g. |-3/2|=3/2. As we know, the field of rational numbers is not an algebraically closed field (e.g. i=-1). Let ¯ be a fixed algebraic closureMathworldPlanetmath of . The absolute value in extends uniquely to ¯. However, ¯ is not completePlanetmathPlanetmath with respect to || (e.g. e=n01/n!¯ because e is transcendental). The completion of ¯ with respect to || is , the field of complex numbers.

Construction of p

We follow the construction of above to build p. Let p be a prime numberMathworldPlanetmath and let p be the p-adic rationals ( or (p-adic numbers). The p-adics, p, are the completion of with respect to the usual p-adic valuation ( ||p. Thus, we regard (p,||p) as a complete metric space. However, the field p is not algebraically closed (e.g. i=-1p if and only if p1mod4). Let ¯p be a fixed algebraic closure of p. The p-adic valuation ||p extends uniquely to ¯p. However:


The field Q¯p is not complete with respect to ||p.


Let βn be defined as:

βn={e2πi/n, if (n,p)=1;1, otherwise.

One can prove that if we define:


then α¯p, although n=mβnpn0 as m (see [1], p. 48, for details). Thus, ¯p is not complete with respect to ||p. ∎


The field of complex p-adic numbers is defined to be the completion of Q¯p with respect to the p-adic absolute value ||p.

Proposition (Properties of Cp).

The field Cp enjoys the following properties:

  1. 1.

    p is algebraically closed.

  2. 2.

    The absolute value ||p extends uniquely to p, which becomes an algebraically closed, complete metric space.

  3. 3.
  4. 4.

    ¯p is dense in p.

  5. 5.

    p is isomorphic to as fields, although they are not isomorphic as topological spacesMathworldPlanetmath.


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