In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a complete field is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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 mathema ...
and
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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 real ...
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 form ...
s, and
complete valued field In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size in ...
s (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 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.
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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
, 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 Municipality, municipal government
* "Place", a type of street or road ...
,
. Given an element
expressed by a fraction
the place
measures the
order of vanishing
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if ...
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