About: Algebraic function field

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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.

Attributes	Values
rdfs:label	Funktionenkörper (de) Algebraic function field (en) Corps de fonctions (fr) 代数函数体 (ja)
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)
dct:subject	Field (mathematics)
Wikipage page ID	447645 (xsd:integer)
Wikipage revision ID	1083939658 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime number theorem Public key cryptography Meromorphic function Morphism (category theory) Algebraic function Holomorphic Riemann surface Inverse Galois problem Quotient ring Complex number Cryptography Mathematics Subring Elliptic curve Equivalence of categories Rational mapping Function field (scheme theory) Function field of an algebraic variety Ideal (ring theory) Topology Drinfeld module Irreducible polynomial Algebraic element Error correcting code Field (mathematics) Finite field Finite field extension Number field Discrete valuation ring Global field Regular map (algebraic geometry) Zariski–Riemann space Surjective Field (mathematics) Absolute value (algebra) Birational geometry Regular scheme Polynomial ring Field extension Field of fractions Glossary of scheme theory Injective function Category (mathematics) Rational number Real number Set (mathematics) Klein surface Scheme (mathematics) Rational functions Transcendence degree Ring homomorphism Morphism of varieties Function field analogy
sameAs	Algebraic function field Algebraic function field Algebraic function field Algebraic function field Algebraic function field Algebraic function field
dbp:wikiPageUsesTemplate	dbt:Math dbt:Refimprove dbt:Reflist dbt:Short_description dbt:Space
has 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)
prov:wasDerivedFrom	wikipedia-en:Algebraic_function_field?oldid=1083939658&ns=0
page length (characters) of wiki page	7303 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Algebraic_function_field
is Link from a Wikipage to another Wikipage of	Algebraic number field Arf invariant Infrastructure (number theory) Number Generalized Riemann hypothesis Christopher Deninger Friedrich Karl Schmidt Cristian Dumitru Popescu Lafforgue's theorem Arithmetic geometry Lonely runner conjecture Lucien Szpiro Function field of an algebraic variety Function field sieve Function field Mason–Stothers theorem Wieferich prime Drinfeld module

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