Rational point

Rational point

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:"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 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.

ee also

*Arithmetic dynamics


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