complex p-adic numbers
First, we review a possible construction of the complex numbers. We start from the rational numbers, , which we consider as a metric space, where the distance is given by the usual absolute value , e.g. . As we know, the field of rational numbers is not an algebraically closed field (e.g. ). Let be a fixed algebraic closure of . The absolute value in extends uniquely to . However, is not complete with respect to (e.g. because e is transcendental). The completion of with respect to is , the field of complex numbers.
We follow the construction of above to build . Let be a prime number and let be the -adic rationals (http://planetmath.org/PAdicIntegers) or (-adic numbers). The -adics, , are the completion of with respect to the usual -adic valuation (http://planetmath.org/PAdicValuation) . Thus, we regard as a complete metric space. However, the field is not algebraically closed (e.g. if and only if ). Let be a fixed algebraic closure of . The -adic valuation extends uniquely to . However:
The field is not complete with respect to .
Let be defined as:
One can prove that if we define:
then , although as (see , p. 48, for details). Thus, is not complete with respect to . ∎
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:
is algebraically closed.
The absolute value extends uniquely to , which becomes an algebraically closed, complete metric space.
is dense in .
- 1 L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.
|Title||complex p-adic numbers|
|Date of creation||2013-03-22 15:13:44|
|Last modified on||2013-03-22 15:13:44|
|Last modified by||alozano (2414)|
|Synonym||complex -adic numbers|