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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