complex p-adic numbers
First, we review a possible construction of the complex numbers. We start from the rational numbers, ℚ, which we consider as a metric space, where the distance is given by the usual absolute value |⋅|, e.g. |-3/2|=3/2. As we know, the field of rational numbers is not an algebraically closed field (e.g. i=√-1∉ℚ). Let ˉℚ be a fixed algebraic closure of ℚ. The absolute value in ℚ extends uniquely to ˉℚ. However, ˉℚ is not complete with respect to |⋅| (e.g. e=∑n≥01/n!∉ˉℚ because e is transcendental). The completion of ˉℚ with respect to |⋅| is ℂ, the field of complex numbers.
Construction of ℂp
We follow the construction of ℂ above to build ℂp. Let p be a prime number and let ℚp be the p-adic rationals (http://planetmath.org/PAdicIntegers) or (p-adic numbers). The p-adics, ℚp, are the completion of ℚ with respect to the usual p-adic valuation (http://planetmath.org/PAdicValuation) |⋅|p. Thus, we regard (ℚp,|⋅|p) as a complete metric space. However, the field ℚp is not algebraically closed (e.g. i=√-1∈ℚp if and only if p≡1mod). Let be a fixed algebraic closure of . The -adic valuation extends uniquely to . However:
Proposition.
The field is not complete with respect to .
Proof.
Let be defined as:
One can prove that if we define:
then , although as (see [1], p. 48, for details). Thus, is not complete with respect to . ∎
Definition.
The field of complex -adic numbers is defined to be the completion of with respect to the -adic absolute value .
Proposition (Properties of ).
The field enjoys the following properties:
-
1.
is algebraically closed.
-
2.
The absolute value extends uniquely to , which becomes an algebraically closed, complete metric space.
- 3.
-
4.
is dense in .
-
5.
is isomorphic to 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 -adic numbers |