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In number theory, a rational point is a point in space each of whose coordinates are rational; that is, the coordinates of the point are elements of the field of rational numbers, as well as being elements of larger fields that contain the rational numbers, such as the real numbers and the complex numbers. For example, is a rational point in 2-dimensional space, since 3 and −67/4 are rational numbers. A special case of a rational point is an integer point, that is, a point all of whose coordinates are integers. E.g., is an integral point in 3-dimensional space. On the other hand, more generally, a ''K''-rational point is a point in a space where each coordinate of the point belongs to the field ''K'', as well as being elements of larger fields containing the field ''K''. This is analogous to rational points, which, as stated above, are contained in fields larger than the rationals. A corresponding special case of ''K''-rational points are those that belong to a ring of algebraic integers existing inside the field ''K''. ==Rational or ''K''-rational points on algebraic varieties== Let ''V'' be an algebraic variety over a field ''K''. When ''V'' is affine, given by a set of equations , with coefficients in ''K'', a ''K''-rational point ''P'' of ''V'' is an ordered n-tuple of numbers from the field ''K'' that is a solution of all of the equations simultaneously. In the general case, a ''K''-rational point of ''V'' is a ''K''-rational point of some affine open subset of ''V''. When ''V'' is projective, defined in some projective space by homogeneous polynomials (with coefficients in ''K''), a ''K''-rational point of ''V'' is a point in the projective space, all of whose coordinates are in ''K'', which is a common solution of all the equations . Sometimes when no confusion is possible (or when ''K'' is the field of the rational numbers), we say rational points instead of ''K''-rational points. Rational (as well as ''K''-rational) points that lie on an algebraic variety (such as an elliptic curve) constitute a major area of current research. For an abelian variety ''A'', the ''K''-rational points form a group. The Mordell-Weil theorem states that the group of rational points of an abelian variety over ''K'' is finitely generated if ''K'' is a number field. The Weil conjectures concern the distribution of rational points on varieties over finite fields, where 'rational points' are taken to mean points from the smallest subfield of the finite field the variety has been defined over. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「rational point」の詳細全文を読む スポンサード リンク
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