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Concept
Rational point
Summary
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 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 K{{sup|n}} of a collection of polynomials with coefficients in k:
:\begin{align}
& f_1(x_1,\ldots,x_n)=0, \
& \qquad \quad \vd
Official source
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