Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Ordered_field> ?p ?o }
- Ordered_field abstract "In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. In 1926, this grew eventually into the Artin–Schreier theory of ordered fields and formally real fields.An ordered field necessarily has characteristic 0, all natural numbers, i.e. the elements 0, 1, 1 + 1, 1 + 1 + 1, ... are distinct. This implies that an ordered field necessarily contains an infinite number of elements: a finite field cannot be ordered.Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any Dedekind-complete ordered field is isomorphic to the real numbers. Squares are necessarily non-negative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is −1. Every ordered field is a formally real field.".
- Ordered_field wikiPageID "22430".
- Ordered_field wikiPageLength "11707".
- Ordered_field wikiPageOutDegree "87".
- Ordered_field wikiPageRevisionID "675186972".
- Ordered_field wikiPageWikiLink Algebraic_number.
- Ordered_field wikiPageWikiLink Algebraic_numbers.
- Ordered_field wikiPageWikiLink American_Mathematical_Society.
- Ordered_field wikiPageWikiLink Archimedean_field.
- Ordered_field wikiPageWikiLink Archimedean_property.
- Ordered_field wikiPageWikiLink Artin–Schreier_theory.
- Ordered_field wikiPageWikiLink Axiomatic_system.
- Ordered_field wikiPageWikiLink Axiomatization.
- Ordered_field wikiPageWikiLink Binary_predicate.
- Ordered_field wikiPageWikiLink Binary_relation.
- Ordered_field wikiPageWikiLink Boolean_space.
- Ordered_field wikiPageWikiLink Category:Ordered_algebraic_structures.
- Ordered_field wikiPageWikiLink Category:Ordered_groups.
- Ordered_field wikiPageWikiLink Category:Real_algebraic_geometry.
- Ordered_field wikiPageWikiLink Characteristic_(algebra).
- Ordered_field wikiPageWikiLink Class_(set_theory).
- Ordered_field wikiPageWikiLink Compact_space.
- Ordered_field wikiPageWikiLink Complex_number.
- Ordered_field wikiPageWikiLink Computable_number.
- Ordered_field wikiPageWikiLink Continuous_function.
- Ordered_field wikiPageWikiLink Continuous_function_(topology).
- Ordered_field wikiPageWikiLink Convex_analysis.
- Ordered_field wikiPageWikiLink David_Hilbert.
- Ordered_field wikiPageWikiLink Dedekind-complete.
- Ordered_field wikiPageWikiLink Discrete_space.
- Ordered_field wikiPageWikiLink Discrete_topology.
- Ordered_field wikiPageWikiLink Examples_of_vector_spaces.
- Ordered_field wikiPageWikiLink Field_(mathematics).
- Ordered_field wikiPageWikiLink Field_extension.
- Ordered_field wikiPageWikiLink Field_of_sets.
- Ordered_field wikiPageWikiLink Finite_field.
- Ordered_field wikiPageWikiLink Formal_power_series.
- Ordered_field wikiPageWikiLink Formally_real_field.
- Ordered_field wikiPageWikiLink Graduate_Studies_in_Mathematics.
- Ordered_field wikiPageWikiLink Hans_Hahn_(mathematician).
- Ordered_field wikiPageWikiLink Hausdorff_space.
- Ordered_field wikiPageWikiLink Hyperreal_number.
- Ordered_field wikiPageWikiLink Hyperreal_numbers.
- Ordered_field wikiPageWikiLink Imaginary_unit.
- Ordered_field wikiPageWikiLink Infimum_and_supremum.
- Ordered_field wikiPageWikiLink Infinitesimal.
- Ordered_field wikiPageWikiLink Inner_product_space.
- Ordered_field wikiPageWikiLink Isomorphism.
- Ordered_field wikiPageWikiLink Least-upper-bound_property.
- Ordered_field wikiPageWikiLink Least_upper_bound.
- Ordered_field wikiPageWikiLink Mathematics.
- Ordered_field wikiPageWikiLink Multiplication.
- Ordered_field wikiPageWikiLink Natural_number.
- Ordered_field wikiPageWikiLink Natural_numbers.
- Ordered_field wikiPageWikiLink Non-Archimedean_ordered_field.
- Ordered_field wikiPageWikiLink Order_topology.
- Ordered_field wikiPageWikiLink Ordered_ring.
- Ordered_field wikiPageWikiLink Ordered_vector_space.
- Ordered_field wikiPageWikiLink Orientation_(vector_space).
- Ordered_field wikiPageWikiLink Otto_Hölder.
- Ordered_field wikiPageWikiLink P-adic_number.
- Ordered_field wikiPageWikiLink P-adic_numbers.
- Ordered_field wikiPageWikiLink Polynomial.
- Ordered_field wikiPageWikiLink Preorder_field.
- Ordered_field wikiPageWikiLink Product_topology.
- Ordered_field wikiPageWikiLink Rational_function.
- Ordered_field wikiPageWikiLink Rational_functions.
- Ordered_field wikiPageWikiLink Rational_number.
- Ordered_field wikiPageWikiLink Real_closed_field.
- Ordered_field wikiPageWikiLink Real_coordinate_space.
- Ordered_field wikiPageWikiLink Real_number.
- Ordered_field wikiPageWikiLink Set_(mathematics).
- Ordered_field wikiPageWikiLink Square_(algebra).
- Ordered_field wikiPageWikiLink Stones_representation_theorem_for_Boolean_algebras.
- Ordered_field wikiPageWikiLink Subbase.
- Ordered_field wikiPageWikiLink Subbasis.
- Ordered_field wikiPageWikiLink Subgroup.
- Ordered_field wikiPageWikiLink Subset.
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- Ordered_field wikiPageWikiLink Superreal_number.
- Ordered_field wikiPageWikiLink Surreal_number.
- Ordered_field wikiPageWikiLink Surreal_numbers.
- Ordered_field wikiPageWikiLink Topological_field.
- Ordered_field wikiPageWikiLink Topological_ring.
- Ordered_field wikiPageWikiLink Total_order.
- Ordered_field wikiPageWikiLink Totally_disconnected_space.
- Ordered_field wikiPageWikiLink Transitive_property.
- Ordered_field wikiPageWikiLink Transitive_relation.
- Ordered_field wikiPageWikiLink Vector_space.
- Ordered_field wikiPageWikiLink Zorns_lemma.
- Ordered_field wikiPageWikiLink File:Invariance_of_less-than-relation_by_multiplication_with_positive_number.svg.
- Ordered_field wikiPageWikiLink File:Translation_invariance_of_less-than-relation.svg.
- Ordered_field wikiPageWikiLinkText "2 definitions".
- Ordered_field wikiPageWikiLinkText "Ordered field".
- Ordered_field wikiPageWikiLinkText "Ordered field#Fan".
- Ordered_field wikiPageWikiLinkText "Ordered field#Harrison topology".
- Ordered_field wikiPageWikiLinkText "Ordered field#Positive cone".
- Ordered_field wikiPageWikiLinkText "Ordered field#Superordered fields".
- Ordered_field wikiPageWikiLinkText "field is ordered".
- Ordered_field wikiPageWikiLinkText "ordered field".