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Concept
Complete field
Résumé
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 valuewhere 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
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.
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