Complete field

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).
Constructions.
Real and complex numbers.
The real numbers are the field with the standard euclidean metric formula_1. Since it is constructed from the completion of formula_2 with respect to this metric, it is a complete field. Extending the reals by its algebraic closure gives the field formula_3 (since its absolute Galois group is formula_4). In this case, formula_3 is also a complete field, but this is not the case in many cases.
p-adic.
The p-adic numbers are constructed from formula_2 by using the p-adic absolute valueformula_7where formula_8. Then using the factorization formula_9 where formula_10 does not divide formula_11, its valuation is the integer formula_12. The completion of formula_2 by formula_14 is the complete field formula_15 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 formula_16.
Function field of a curve.
For the function field formula_17 of a curve formula_18, every point formula_19 corresponds to an absolute value, or place, formula_14. Given an element formula_21 expressed by a fraction formula_22, the place formula_14 measures the order of vanishing of formula_24 at formula_10 minus the order of vanishing of formula_26 at formula_10. Then, the completion of formula_17 at formula_10 gives a new field. For example, if formula_30 at formula_31, the origin in the affine chart formula_32, then the completion of formula_17 at formula_10 is isomorphic to the power-series ring formula_35.