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 numbers).
A field is a set with binary operations and (called addition and multiplication, respectively), along with elements and such that for all , the following relations hold: [1]
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 : [2]
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. [2] [3]
The real numbers are the field with the standard Euclidean metric , and this measure is complete. [2] Extending the reals by adding the imaginary number satisfying gives the field , which is also a complete field. [3]
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 [4]