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 (such as the p-adic numberss).

Definitions

Field

A field is a set with binary operations (called addition and multiplication, respectively), along with elements, the following relations hold:

1. has a solution

Complete metric

A metric is a function and sends them to a non-negative real number:

1. if and only if

A sequence in the space is Cauchy, and a metric is then complete if every Cauchy sequence in the metric space converges. Every convergent sequence is Cauchy, however the converse does not hold in general.

Constructions

Real and complex numbers

, and this measure is complete. Extending the reals by adding the imaginary number, which is also a complete field.

p-adic

by using the p-adic absolute value

References

See also