In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers).
The real numbers are the field with the standard Euclidean metric . Since it is constructed from the completion of with respect to this metric, it is a complete field. Extending the reals by its algebraic closure gives the field (since its absolute Galois group is ). In this case, is also a complete field, but this is not the case in many cases.
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[1] is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted
For the function field of a curve every point corresponds to an absolute value, or place, . Given an element expressed by a fraction the place measures the order of vanishing of at minus the order of vanishing of at Then, the completion of at gives a new field. For example, if at the origin in the affine chart then the completion of at is isomorphic to the power-series ring