Rational point

A point on an algebraic variety whose coordinates belong to a specified base field. Over the rational numbers, its coordinates are rational numbers.

Algebraic variety: A geometric object defined by polynomial equations over a field. Rational points are points of varieties whose coordinates satisfy the defining equations.
Coordinate ring: A ring of polynomial functions on an algebraic variety. Its equations encode the constraints that candidate coordinate tuples must satisfy.
Diophantine equation: A polynomial equation whose solutions are sought in specified number systems, often the integers or rationals. Its rational solutions are rational points on the variety defined by the equation.
Algebraic point: A point whose coordinates lie in an algebraic extension of the base field. Every rational point is algebraic, but algebraic points may require a larger field.
Field (mathematics): A set with addition, subtraction, multiplication, and division by nonzero elements satisfying familiar arithmetic laws. The chosen field determines which coordinates count as rational.
Homogeneous coordinates: Coordinates defined up to common nonzero scaling, commonly used for projective points. They describe projective rational points without privileging an affine chart.
Mordell–Weil theorem: The theorem that an abelian variety over a number field has a finitely generated group of rational points. It gives a structural finiteness result for rational points on abelian varieties.
Integer point: A solution whose coordinates are integers in a chosen coordinate model. Integer points impose integrality, a stronger condition than rationality in that model.
Polynomial equation: An equation formed by setting a polynomial expression equal to another expression, usually zero. Its solutions supply the points whose coordinates are tested for field membership.
Base change: The process of extending or changing the field of scalars in an algebraic construction. Changing the base field changes which points qualify as rational.

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