Rational point
Encyclopedia
In number theory
, a K-rational point is a point on an algebraic variety
where each coordinate of the point belongs to the field K. This means that, if the variety is given by a set of equations
then the K-rational points are solutions (x1, ..., xn) ∈Kn of the equations. In the parlance of morphisms of schemes, a K-rational point of a scheme X is just a morphism Spec K → X. The set of K-rational points is usually denoted X(K).
If a scheme
or variety X is defined over a field
k, a point x ∈ X is also called rational point if its residue field
k(x) is isomorphic to k.
Rational points of varieties 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.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a K-rational point is a point on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
where each coordinate of the point belongs to the field K. This means that, if the variety is given by a set of equations
- fi(x1, ..., xn)=0, j=1, ..., m
then the K-rational points are solutions (x1, ..., xn) ∈Kn of the equations. In the parlance of morphisms of schemes, a K-rational point of a scheme X is just a morphism Spec K → X. The set of K-rational points is usually denoted X(K).
If a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
or variety X is defined over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
k, a point x ∈ X is also called rational point if its residue field
Residue field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field...
k(x) is isomorphic to k.
Rational points of varieties constitute a major area of current research.
For an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
A, the K-rational points form a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. 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
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
concern the distribution of rational points on varieties over finite fields.