# Complete field

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In [mathematics](/source/mathematics), a '''complete field''' is a [field](/source/Field_(mathematics)) equipped with a [metric](/source/Metric_(mathematics)) and [complete](/source/Complete_metric_space) with respect to that metric. A field supports the elementary operations of [addition](/source/addition), [subtraction](/source/subtraction), [multiplication](/source/multiplication), and [division](/source/Division_(mathematics)), while a metric represents the [distance](/source/distance) between two points in the set. Basic examples include the [real number](/source/real_number)s, the [complex number](/source/complex_number)s, and [complete valued field](/source/complete_valued_field)s (such as the [''p''-adic number](/source/p-adic_numbers)s).

== Definitions ==

=== Field ===

A [field](/source/Field_(mathematics)) is a [set](/source/Set_(mathematics)) <math>F</math> with [binary operations](/source/binary_operations) <math>+</math> and <math>\cdot</math> (called ''addition'' and ''multiplication'', respectively), along with [elements](/source/Element_(mathematics)) <math>0</math> and <math>1</math> such that for all <math>a,b,c \in F</math>, the following relations hold:<ref>{{cite book |last1=Hungerford |first1=Thomas W. |title=Abstract Algebra: an introduction |date=2014 |publisher=Brooks/Cole, Cengage Learning |location=Boston, MA |isbn=978-1-111-56962-4 |pages=44,49 |edition=Third}}</ref>
# <math>a+(b+c)=(a+b)+c</math>
# <math>a+b=b+a</math>
# <math>a+0=a=0+a</math>
# <math>a+x=0</math> has a solution
# <math>a(bc)=(ab)c</math>
# <math>ab=ba</math>
# <math>a(b+c)=ab+ac</math> and <math>(a+b)c=ac+bc</math>
# <math>a1=a=1a</math>
# <math>ax=1</math> has a solution for <math>a \neq 0</math>

=== Complete metric ===

A [metric](/source/Metric_space) on a set <math>F</math> is a [function](/source/Function_(mathematics)) <math>d: F^2 \to [0, \infty)</math>, that is, it takes two points in <math>F</math> and sends them to a [non-negative](/source/non-negative) [real number](/source/real_number), such that the following relations hold for all <math>x,y,z \in F</math>:<ref name="Folland">{{cite book |last1=Folland |first1=Gerald B. |title=Real analysis: modern techniques and their applications |date=1999 |publisher=New York J. Wiley & sons |location=Chichester Weinheim [etc.] |isbn=0-471-31716-0 |pages=13–14 |edition=2nd}}</ref>
# <math>d(x,y) = 0</math> [if and only if](/source/if_and_only_if) <math>x=y</math>
# <math>d(x,y)=d(y,x)</math>
# <math>d(x,y) \leq d(x,z)+d(z,y)</math>
A [sequence](/source/sequence) <math>x_n</math> in the space is [Cauchy](/source/Cauchy_sequence) with respect to this metric if for all <math>\epsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that for all <math>n,m \geq N</math> we have <math>d(x_n,x_m) < \epsilon</math>, and a metric is then [complete](/source/Complete_metric_space) if every Cauchy sequence in the metric space [converges](/source/Convergent_sequence), that is, there is some <math>x \in F</math> where for all <math>\epsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that for all <math>n \geq N</math> we have <math>d(x_n,x) < \epsilon</math>. Every convergent sequence is Cauchy, however the converse does not hold in general.<ref name="Folland" /><ref name="Rudin">{{cite book |last1=Rudin |first1=Walter |title=Principles of mathematical analysis |date=2008 |publisher=McGraw-Hill |location=New York |isbn=978-0-07-054235-8 |pages=47,52–54 |edition=3., [Nachdr.]}}</ref>

==Constructions==

===Real and complex numbers===

The real numbers are the field with the standard Euclidean metric <math>|x-y|</math>, and this measure is complete.<ref name="Folland" /> Extending the reals by adding the [imaginary number](/source/imaginary_number) <math>i</math> satisfying <math>i^2=-1</math> gives the field <math>\Complex</math>, which is also a complete field.<ref name="Rudin" />

===p-adic===

The p-adic numbers are constructed from <math>\Q</math> by using the p-adic absolute value<blockquote><math>v_p(a/b) = v_p(a) - v_p(b)</math></blockquote>where <math>a,b \in \Z.</math> Then using the factorization <math>a = p^nc</math> where <math>p</math> does not divide <math>c,</math> its valuation is the integer <math>n</math>. The completion of <math>\Q</math> by <math>v_p</math> is the complete field <math>\Q_p</math> 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 <math>\Complex_p.</math><ref>{{Cite book|last=Koblitz, Neal.|title=P-adic Numbers, p-adic Analysis, and Zeta-Functions|date=1984|publisher=Springer New York|isbn=978-1-4612-1112-9|edition= Second|location=New York, NY|pages=52–75|oclc=853269675}}</ref>

==References==

{{reflist}}

==See also==

* {{annotated link|Completion (algebra)}}
* {{annotated link|Complete topological vector space}}
* {{annotated link|Hensel's lemma}}
* {{annotated link|Henselian ring}}
* {{annotated link|Compact group}}
* {{annotated link|Locally compact field}}
* {{annotated link|Locally compact quantum group}}
* {{annotated link|Locally compact group}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Ostrowski's theorem}}
* {{annotated link|Topological abelian group}}
* {{annotated link|Topological field}}
* {{annotated link|Topological group}}
* {{annotated link|Topological module}}
* {{annotated link|Topological ring}}
* {{annotated link|Topological semigroup}}
* {{annotated link|Topological vector space}}

Category:Field theory
Category:Topological algebra

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