Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Field_(mathematics)> ?p ?o }
- Field_(mathematics) abstract "In abstract algebra, a field is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, algebraic number fields, p-adic fields, and so forth.Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry.As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fields allow for division (though not division by zero), while a ring need not possess multiplicative inverses; for example the integers form a ring, but 2x = 1 has no solution in integers. Also, the multiplication operation in a field is required to be commutative. A ring in which division is possible but commutativity is not assumed (such as the quaternions) is called a division ring or skew field. (Historically, division rings were sometimes referred to as fields, while fields were called commutative fields.)As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following (not exhaustive) chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields".
- Field_(mathematics) wikiPageExternalLink fields.htm.
- Field_(mathematics) wikiPageExternalLink index.php.
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- Field_(mathematics) wikiPageRevisionID "678421230".
- Field_(mathematics) wikiPageWikiLink Abelian_group.
- Field_(mathematics) wikiPageWikiLink Abel–Ruffini_theorem.
- Field_(mathematics) wikiPageWikiLink Absolute_Galois_group.
- Field_(mathematics) wikiPageWikiLink Abstract_algebra.
- Field_(mathematics) wikiPageWikiLink Addition.
- Field_(mathematics) wikiPageWikiLink Additive_inverse.
- Field_(mathematics) wikiPageWikiLink Algebraic_closure.
- Field_(mathematics) wikiPageWikiLink Algebraic_extension.
- Field_(mathematics) wikiPageWikiLink Algebraic_geometry.
- Field_(mathematics) wikiPageWikiLink Algebraic_geometry_and_analytic_geometry.
- Field_(mathematics) wikiPageWikiLink Algebraic_number.
- Field_(mathematics) wikiPageWikiLink Algebraic_number_field.
- Field_(mathematics) wikiPageWikiLink Algebraic_structure.
- Field_(mathematics) wikiPageWikiLink Algebraic_variety.
- Field_(mathematics) wikiPageWikiLink Algebraically_closed.
- Field_(mathematics) wikiPageWikiLink Algebraically_closed_field.
- Field_(mathematics) wikiPageWikiLink Angle_trisection.
- Field_(mathematics) wikiPageWikiLink Associative_property.
- Field_(mathematics) wikiPageWikiLink Associativity.
- Field_(mathematics) wikiPageWikiLink Automorphism.
- Field_(mathematics) wikiPageWikiLink Axiom_of_choice.
- Field_(mathematics) wikiPageWikiLink Bijection.
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- Field_(mathematics) wikiPageWikiLink Binary_relation.
- Field_(mathematics) wikiPageWikiLink Calculus.
- Field_(mathematics) wikiPageWikiLink Cambridge_University_Press.
- Field_(mathematics) wikiPageWikiLink Cardinal_number.
- Field_(mathematics) wikiPageWikiLink Category:Algebraic_structures.
- Field_(mathematics) wikiPageWikiLink Category:Field_theory.
- Field_(mathematics) wikiPageWikiLink Category_of_fields.
- Field_(mathematics) wikiPageWikiLink Category_of_rings.
- Field_(mathematics) wikiPageWikiLink Category_theory.
- Field_(mathematics) wikiPageWikiLink Characteristic_(algebra).
- Field_(mathematics) wikiPageWikiLink Class_(set_theory).
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- Field_(mathematics) wikiPageWikiLink Coding_theory.
- Field_(mathematics) wikiPageWikiLink Coefficient.
- Field_(mathematics) wikiPageWikiLink Combinatorics.
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- Field_(mathematics) wikiPageWikiLink Commutative_ring.
- Field_(mathematics) wikiPageWikiLink Commutativity.
- Field_(mathematics) wikiPageWikiLink Compact_space.
- Field_(mathematics) wikiPageWikiLink Compact_topological_space.
- Field_(mathematics) wikiPageWikiLink Compass.
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- Field_(mathematics) wikiPageWikiLink Compass_and_straightedge.
- Field_(mathematics) wikiPageWikiLink Complete_field.
- Field_(mathematics) wikiPageWikiLink Complete_metric_space.
- Field_(mathematics) wikiPageWikiLink Completion_(metric_space).
- Field_(mathematics) wikiPageWikiLink Complex_number.
- Field_(mathematics) wikiPageWikiLink Computable_number.
- Field_(mathematics) wikiPageWikiLink Computer_science.
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- Field_(mathematics) wikiPageWikiLink Constructible_numbers.
- Field_(mathematics) wikiPageWikiLink Continuous_function.
- Field_(mathematics) wikiPageWikiLink Cryptography.
- Field_(mathematics) wikiPageWikiLink Cyclic_group.
- Field_(mathematics) wikiPageWikiLink David_Hilbert.
- Field_(mathematics) wikiPageWikiLink Derivation_(abstract_algebra).
- Field_(mathematics) wikiPageWikiLink Derivation_(differential_algebra).
- Field_(mathematics) wikiPageWikiLink Differential_Galois_theory.
- Field_(mathematics) wikiPageWikiLink Differential_algebra.
- Field_(mathematics) wikiPageWikiLink Differential_field.
- Field_(mathematics) wikiPageWikiLink Dimension_(vector_space).
- Field_(mathematics) wikiPageWikiLink Distributive_law.
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- Field_(mathematics) wikiPageWikiLink Distributivity.
- Field_(mathematics) wikiPageWikiLink Division_(mathematics).
- Field_(mathematics) wikiPageWikiLink Division_by_zero.
- Field_(mathematics) wikiPageWikiLink Division_ring.
- Field_(mathematics) wikiPageWikiLink Domain_(ring_theory).
- Field_(mathematics) wikiPageWikiLink E._H._Moore.
- Field_(mathematics) wikiPageWikiLink Elementary_group_theory.
- Field_(mathematics) wikiPageWikiLink Emil_Artin.
- Field_(mathematics) wikiPageWikiLink Ernst_Steinitz.
- Field_(mathematics) wikiPageWikiLink Euclidean_field.
- Field_(mathematics) wikiPageWikiLink Exclusive_or.
- Field_(mathematics) wikiPageWikiLink Exponential_field.
- Field_(mathematics) wikiPageWikiLink Exponential_function.
- Field_(mathematics) wikiPageWikiLink Fermats_Last_Theorem.
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- Field_(mathematics) wikiPageWikiLink Field_automorphism.
- Field_(mathematics) wikiPageWikiLink Field_extension.
- Field_(mathematics) wikiPageWikiLink Field_of_fractions.
- Field_(mathematics) wikiPageWikiLink Field_with_one_element.
- Field_(mathematics) wikiPageWikiLink Finite_extension.
- Field_(mathematics) wikiPageWikiLink Finite_field.
- Field_(mathematics) wikiPageWikiLink First-order_logic.
- Field_(mathematics) wikiPageWikiLink First-order_sentence.
- Field_(mathematics) wikiPageWikiLink Formal_power_series.
- Field_(mathematics) wikiPageWikiLink Foundations_of_mathematics.