Algebraic function field (original) (raw)
In mathematics, an algebraic function field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K = k(x1,...,xn) of rational functions in n variables over k.
| Property | Value |
|---|---|
| dbo:abstract | In mathematics, an algebraic function field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K = k(x1,...,xn) of rational functions in n variables over k. (en) (Algebraische) Funktionenkörper sind in der Mathematik algebraische Entsprechungen geometrischer Objekte. Funktionenkörper über endlichen Körpern spielen auch in der algebraischen Zahlentheorie eine wichtige Rolle. (de) En mathématiques, un corps de fonctions est un corps commutatif F de type fini sur un corps de base K. On le note habituellement F/K, ou, si le contexte est clair, seulement F. De façon équivalente un corps de fonctions « à n variables » est une extension finie F d'un corps K(t1, … , tn) de fractions rationnelles à n indéterminées. F est alors de degré de transcendance n sur K. * Une extension L de k est un corps de fonctions (à n variables) si et seulement si c'est le (en) d'une variété algébrique intègre sur k (de dimension n). * Un corps de fonctions à une variable sur un corps fini est un corps global de caractéristique positive. C'est le corps des fonctions rationnelles d'une courbe projective lisse intègre sur un corps fini. (fr) 数学では、体 k 上の n 変数の代数函数体 (algebraic function field)（単に、函数体とも言う）は、k 上に超越次数 n を持つ有限生成な体の拡大 K/k である。同じことであるが、k 上の n 変数の代数函数体は、k 上の n 変数の有理函数の体 k(x1, ..., xn) の有限拡大として定義できる。 (ja) |
| dbo:wikiPageID | 447645 (xsd:integer) |
| dbo:wikiPageLength | 7303 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1083939658 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Prime_number_theorem dbr:Public_key_cryptography dbr:Meromorphic_function dbr:Morphism_(category_theory) dbr:Algebraic_function dbr:Holomorphic dbr:Riemann_surface dbr:Inverse_Galois_problem dbr:Quotient_ring dbr:Complex_number dbr:Cryptography dbr:Mathematics dbr:Subring dbr:Elliptic_curve dbr:Equivalence_of_categories dbr:Rational_mapping dbr:Function_field_(scheme_theory) dbr:Function_field_of_an_algebraic_variety dbr:Ideal_(ring_theory) dbr:Topology dbr:Drinfeld_module dbr:Irreducible_polynomial dbr:Algebraic_element dbr:Error_correcting_code dbr:Field_(mathematics) dbr:Finite_field dbr:Finite_field_extension dbr:Number_field dbr:Discrete_valuation_ring dbr:Global_field dbr:Regular_map_(algebraic_geometry) dbr:Zariski–Riemann_space dbr:Surjective dbc:Field_(mathematics) dbr:Absolute_value_(algebra) dbr:Birational_geometry dbr:Regular_scheme dbr:Polynomial_ring dbr:Field_extension dbr:Field_of_fractions dbr:Glossary_of_scheme_theory dbr:Injective_function dbr:Category_(mathematics) dbr:Rational_number dbr:Real_number dbr:Set_(mathematics) dbr:Klein_surface dbr:Scheme_(mathematics) dbr:Rational_functions dbr:Transcendence_degree dbr:Ring_homomorphism dbr:Morphism_of_varieties dbr:Function_field_analogy |
| dbp:wikiPageUsesTemplate | dbt:Math dbt:Refimprove dbt:Reflist dbt:Short_description dbt:Space |
| dct:subject | dbc:Field_(mathematics) |
| rdfs:comment | In mathematics, an algebraic function field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K = k(x1,...,xn) of rational functions in n variables over k. (en) (Algebraische) Funktionenkörper sind in der Mathematik algebraische Entsprechungen geometrischer Objekte. Funktionenkörper über endlichen Körpern spielen auch in der algebraischen Zahlentheorie eine wichtige Rolle. (de) 数学では、体 k 上の n 変数の代数函数体 (algebraic function field)（単に、函数体とも言う）は、k 上に超越次数 n を持つ有限生成な体の拡大 K/k である。同じことであるが、k 上の n 変数の代数函数体は、k 上の n 変数の有理函数の体 k(x1, ..., xn) の有限拡大として定義できる。 (ja) En mathématiques, un corps de fonctions est un corps commutatif F de type fini sur un corps de base K. On le note habituellement F/K, ou, si le contexte est clair, seulement F. De façon équivalente un corps de fonctions « à n variables » est une extension finie F d'un corps K(t1, … , tn) de fractions rationnelles à n indéterminées. F est alors de degré de transcendance n sur K. (fr) |
| rdfs:label | Funktionenkörper (de) Algebraic function field (en) Corps de fonctions (fr) 代数函数体 (ja) |
| owl:sameAs | freebase:Algebraic function field wikidata:Algebraic function field dbpedia-de:Algebraic function field dbpedia-fr:Algebraic function field dbpedia-ja:Algebraic function field https://global.dbpedia.org/id/4Nrov |
| prov:wasDerivedFrom | wikipedia-en:Algebraic_function_field?oldid=1083939658&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Algebraic_function_field |
| is dbo:wikiPageWikiLink of | dbr:Algebraic_number_field dbr:Arf_invariant dbr:Infrastructure_(number_theory) dbr:Number dbr:Generalized_Riemann_hypothesis dbr:Christopher_Deninger dbr:Friedrich_Karl_Schmidt dbr:Cristian_Dumitru_Popescu dbr:Lafforgue's_theorem dbr:Arithmetic_geometry dbr:Lonely_runner_conjecture dbr:Lucien_Szpiro dbr:Function_field_of_an_algebraic_variety dbr:Function_field_sieve dbr:Function_field dbr:Mason–Stothers_theorem dbr:Wieferich_prime dbr:Drinfeld_module dbr:K-groups_of_a_field dbr:Algebraic_curve dbr:Algebraic_geometry dbr:Algebraic_number_theory dbr:Field_(mathematics) dbr:Number_theory dbr:Quasi-algebraically_closed_field dbr:Abhyankar's_conjecture dbr:Abstract_algebra dbr:Artin–Schreier_curve dbr:Maple_(software) dbr:Kunihiko_Kodaira dbr:Klein_surface dbr:Peter_Roquette dbr:Finiteness_properties_of_groups dbr:Superelliptic_curve dbr:Tamagawa_number |
| is foaf:primaryTopic of | wikipedia-en:Algebraic_function_field |