Field theory and polynomials Concept
Complete ordered field
Summary
A complete ordered field is an ordered field in which each nonempty subset that is bounded above has a supremum in . The real numbers form a complete ordered field, by the completeness axiom. Any two complete ordered fields are isomorphic, by a unique isomorphism that preserves the order. Thus "the complete ordered field" is a characterization of .
Remarks
The ordered field is not complete. Each complete ordered field is Archimedean. Other forms of completeness of are equivalent for Archimedean ordered fields, for example the convergence of each Cauchy sequence.
Where this leads
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Further reading
- Web https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers, Forms of completeness
- Web https://en.wikipedia.org/wiki/Real_number, "The complete ordered field"
About this page
| Tag | 0A9E6X0S3E |
|---|---|
| URI | m2://0A9E6X0S3E |
| Kind | concept |
| Writing | W0 stub |
| MSC | 12J15 |
| Specializes | Archimedean ordered field |
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