In mathematics, a '''complete field''' is a field equipped with a metric and complete with respect to that metric. A field supports the elementary operations of addition, subtraction, multiplication, and division, while a metric represents the distance between two points in the set. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the ''p''-adic numbers).
== Definitions ==
=== Field ===
A field is a set <math>F</math> with binary operations <math>+</math> and <math>\cdot</math> (called ''addition'' and ''multiplication'', respectively), along with elements <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 on a set <math>F</math> is a function <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 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 <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 <math>x_n</math> in the space is Cauchy 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 if every Cauchy sequence in the metric space converges, 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 <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