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 x n + y n = 1 {\displaystyle x^{n}+y^{n}=1} has no other rational points than (1, 0), (0, 1), and, if n is even, (–1, 0) and (0, –1).
Definition Given a field k, and an algebraically closed extension K of k, an affine variety X over k is the set of common zeros in Kn of a collection of polynomials with coefficients in k:
f 1 ( x 1 , … , x n ) = 0 , ⋮ f r ( x 1 , … , x n ) = 0. {\displaystyle {\begin{aligned}&f_{1}(x_{1},\ldots ,x_{n})=0,\\&\qquad \quad \vdots \\&f_{r}(x_{1},\dots ,x_{n})=0.\end{aligned}}}
These common zeros are called the points of X. A k-rational point (or k-point) of X is a point of X that belongs to kn, that is, a sequence ( a 1 , … , a n ) {\displaystyle (a_{1},\dots ,a_{n})} of n elements of k such that f j ( a 1 , … , a n ) = 0 {\displaystyle f_{j}(a_{1},\dots ,a_{n})=0} for all j. The set of k-rational points of X is often denoted X(k). Sometimes, when the field k is understood, or when k is the field Q {\displaystyle \mathbb {Q} } of rational numbers, one says "rational point" instead of "k-rational point". For example, the rational points of the unit circle of equation
x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1}
are the pairs of rational numbers
( a c , b c ) , {\displaystyle \left({\frac {a}{c}},{\frac {b}{c}}\right),}
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