Complete field
In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, 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 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[1] 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, . 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
- ^ Koblitz, Neal. (1984). P-adic Numbers, p-adic Analysis, and Zeta-Functions (Second ed.). New York, NY: Springer New York. pp. 52–75. ISBN 978-1-4612-1112-9. OCLC 853269675.
See also
- Completion (algebra) – in algebra, any of several related functors on rings and modules that result in complete topological rings and modulesPages displaying wikidata descriptions as a fallback
- Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
- Hensel's lemma – Result in modular arithmetic
- Henselian ring – local ring in which Hensel’s lemma holdsPages displaying wikidata descriptions as a fallback
- Compact group – Topological group with compact topology
- Locally compact field
- Locally compact quantum group – relatively new C*-algebraic approach toward quantum groupsPages displaying wikidata descriptions as a fallback
- Locally compact group – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be definedPages displaying wikidata descriptions as a fallback
- Ordered topological vector space
- Ostrowski's theorem – On all absolute values of rational numbers
- Topological abelian group – topological group whose group is abelianPages displaying wikidata descriptions as a fallback
- Topological field – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets
- Topological group – Group that is a topological space with continuous group action
- Topological module
- Topological ring – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback
- Topological semigroup – semigroup with continuous operationPages displaying wikidata descriptions as a fallback
- Topological vector space – Vector space with a notion of nearness
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