Complete field


In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. A field supports the elementary operations of addition, subtraction, multiplication, and division, while a metric represents the distance between two points in the set. Basic examples include the real numbers, the complex numbers, and complete valued fields.

Definitions

Field

A field is a set with binary operations and , along with elements and such that for all, the following relations hold:
  4. has a solution
  7. and
  9. has a solution for

    Complete metric

A metric on a set is a function, that is, it takes two points in and sends them to a non-negative real number, such that the following relations hold for all :
  1. if and only if
A sequence in the space is Cauchy with respect to this metric if for all there exists an such that for all we have, and a metric is then complete if every Cauchy sequence in the metric space converges, that is, there is some where for all there exists an such that for all we have. Every convergent sequence is Cauchy, however the converse does not hold in general.

Constructions

Real and complex numbers

The real numbers are the field with the standard Euclidean metric, and this measure is complete. Extending the reals by adding the imaginary number satisfying gives the field, which is also a complete field.

p-adic

The p-adic numbers are constructed from by using the p-adic absolute value
where Then using the factorization where does not divide its valuation is the integer. The completion of by is the complete field called the p-adic numbers. This is a case where the field is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted