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).

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

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

its valuation is the integer

The completion of

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

corresponds to an absolute value, or place,

Given an element

expressed by a fraction

measures the order of vanishing of

minus the order of vanishing of

gives a new field.

the origin in the affine chart

is isomorphic to the power-series ring

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