Group of rational points on the unit circle

The group of rational points on the unit circle consists of the points with rational coordinates, applies. The amount of these points is closely related to the prime Pythagorean triples. Is a primitive right triangle given with integer coprime side lengths, the hypotenuse is, then there is on the unit circle the rational point. Conversely, if a rational point on the unit circle, then there is a primitive right triangle with the sides, the least common multiple of the denominators of and is.

Group operation

The set of rational points forms an infinite Abelian group. The neutral element is the point. The group operation or " sum " is. Geometrically, this is the Winkeladditon when and where the angle of the radius vector with the radius vector in the mathematically positive sense. Thus, if each and form with the bracket and whose sum is the rational point on the unit circle at the angle in the sense of the ordinary addition of angles.

One identifies each point by the complex number, so the corresponding addition in the multiplication.

Group structure

The group is isomorphic to an infinite direct sum of cyclic subgroups of:

Is generated by the sub-group, and which are those subgroups, which are generated from points of the shape.

Swell

• Ernest J. Eckert: The Group of Primitive Pythagorean Triangles. Mathematics Magazine Vol 57 No. 1 ( January, 1984), pp. 22-26

• Group (mathematics)
• Group Theory