About: Rational point

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In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for n > 2, the Fermat curve of equation has no other rational points than (1, 0), (0, 1), and, if n is even, (–1, 0) and (0, –1).

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rdf:type	place
rdfs:label	Point rationnel (fr) 有理点 (ja) 유리점 (ko) Rational point (en)
rdfs:comment	En théorie des nombres et géométrie algébrique, les points rationnels d'une variété algébrique définie sur un corps sont, lorsque X est définie par un système d'équations polynomiales, les solutions dans k de ce système. (fr) In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for n > 2, the Fermat curve of equation has no other rational points than (1, 0), (0, 1), and, if n is even, (–1, 0) and (0, –1). (en) 数論において有理点（ゆうりてん、英: rational point）とは、各座標の値が全て有理数であるような空間の点のことである。例えば、点 (3, −67/4) は 3 も −67/4 も有理数であるため、2次元空間内の有理点である。有理点の特別な場合として(integer point)があり、これは座標値が全て整数の点である。例えば、(1, −5, 0) は 3次元空間内の整数点である。より一般に K を任意の体とするとき、K-有理点は点の各々の座標値が体 K に属するような点と定義される。同様に、特別な場合である K-整数点は、各座標値が数体 K 内の代数的整数の環の元である点と定義される。 (ja) 대수적 수론과 대수기하학에서, 대수다양체 또는 스킴의 유리점(有理點, 영어: rational point)은 좌표가 모두 유리수인 점이다. (ko)
dcterms:subject	Diophantine geometry
Wikipage page ID	2960200 (xsd:integer)
Wikipage revision ID	1086937865 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Projective space Pythagorean triple Roger Heath-Brown Mordell–Weil theorem Morphism of schemes Rational variety Beniamino Segre Algebraic group Algebraically closed field Algorithm Annales de l'Institut Fourier Richard Taylor (mathematician) Cubic surface Unit circle Inventiones Mathematicae Real point Complex number Chevalley–Warning theorem Genus (mathematics) Christopher Hooley Elliptic curve Enrico Bombieri Andrew Wiles André Weil Commutative ring Yuri Manin Étale cohomology Functor Functor represented by a scheme Spectrum of a ring Hasse principle Hasse–Minkowski theorem Affine variety Algebraic curve Algebraic geometry Algebraic variety American Mathematical Society Ernst Sejersted Selmer Faltings's theorem Fermat's Last Theorem Field (mathematics) Finite field Diophantine geometry Number field Number theory Diophantine geometry Fano variety Kobayashi metric Projective cone Projective variety Hardy–Littlewood circle method Height function J. W. S. Cassels Jean-Louis Colliot-Thélène Tate–Shafarevich group Hypersurface Hélène Esnault Arithmetic dynamics Abelian group Abelian variety János Kollár K3 surface Birational geometry Birch's theorem Homogeneous polynomial Zero of a function Diophantine equation Associative algebra Pierre Deligne Springer Nature Fermat curve Fiber product of schemes Field extension Integer Algebraically closed extension Orbifold Rational number Rationally connected variety Real number Serge Lang Scheme (mathematics) Section (category theory) Finitely generated abelian group Manin conjecture Smooth scheme Perfect field P-adic field

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