In
mathematics, a complete field is a
field equipped with a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
and
complete with respect to that metric. Basic examples include the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, 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
. 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
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
gives the field
(since its
absolute Galois group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' t ...
is
). In this case,
is also a complete field, but this is not the case in many cases.
p-adic
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
is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted
Function field of a curve
For the function field
of a curve
every point
corresponds to an
absolute value, or
place
Place may refer to:
Geography
* Place (United States Census Bureau), defined as any concentration of population
** Census-designated place, a populated area lacking its own municipal government
* "Place", a type of street or road name
** Often ...
,
. 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
References
See also
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Field (mathematics)
{{Abstract-algebra-stub