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In [[mathematics]], a '''complete field''' is a [[Field (mathematics)|field]] equipped with a [[Metric (mathematics)|metric]] and [[Complete metric space|complete]] with respect to that metric. Basic examples include the [[real number]]s, the [[complex number]]s, and [[complete valued field]]s (such as the [[p-adic numbers|''p''-adic number]]s).

==Constructions==

===Real and complex numbers===

The real numbers are the field with the standard Euclidean metric <math>|x-y|</math>. Since it is constructed from the completion of <math>\Q</math> with respect to this metric, it is a complete field. Extending the reals by its [[algebraic closure]] gives the field <math>\Complex</math> (since its [[absolute Galois group]] is <math>\Z/2</math>). In this case, <math>\Complex</math> is also a complete field, but this is not the case in many cases.

===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<ref>{{Cite book|last=Koblitz, Neal.|url=https://www.worldcat.org/oclc/853269675|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> 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>

===Function field of a curve===

For the function field <math>k(X)</math> of a curve <math>X/k,</math> every point <math>p \in X</math> corresponds to an [[Absolute value (algebra)|absolute value]], or [[Place of a field|place]], <math>v_p</math>. Given an element <math>f \in k(X)</math> expressed by a fraction <math>g/h,</math> the place <math>v_p</math> measures the [[Multiplicity (mathematics)|order of vanishing]] of <math>g</math> at <math>p</math> minus the order of vanishing of <math>h</math> at <math>p.</math> Then, the completion of <math>k(X)</math> at <math>p</math> gives a new field. For example, if <math>X = \mathbb{P}^1</math> at <math>p = [0:1],</math> the origin in the affine chart <math>x_1 \neq 0,</math> then the completion of <math>k(X)</math> at <math>p</math> is isomorphic to the power-series ring <math>k((x)).</math>

==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 (mathematics)]]


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