User:Richard L. Peterson/The group of rational points on the unit circle

The rational points on the unit circle are those points (x,y) such both x and y are rational numbers(fractions) and satisfy xx + yy = 1. The set of such points turn out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integral side lengths 'a', 'b', 'c', with 'c' the hypotenuse, such that the sides have no common factor larger than 1. Then on the unit circle there exists the rational point (a/c,b/c). Conversely, if (x,y) is a rational point on the unit circle, then there exists an primitive right triangle with sides 'xc', 'yc', 'c'.

Group operation
The set of rational points forms an infinite abelian group, which shall be called 'G' in this article. The identity element is the point (1,0). The group operation, or "sum" is (x,y) + (t,u) = (xt-uy,xu+yt). This is angle addition since x = cos(A) and y = sin(A), where A is the angle the radius vector (x,y) makes with the radius vector (1,0), measured counter clockwise. So with (x,y) and (t,u) forming angles A and B, respectively, with (1,0), their sum (xt-uy,xu+yt) is just the rational point on the unit circle with angle A + B(ordinary sum).

Group structure
The structure of 'G' is an infinite sum of cyclic groups. If C4 denotes the cyclic subgroup with 4 elements generated by the point (0,1), and Z is any infinite cyclic subgroup generated by a point of form (a/p,b/p) where p is a prime of form 4k + 1, (and 'a', 'b' are positive)then G is isomorphic to C4 + Z + Z + ..., going on forever. Since it is an infinite sum rather than infinite product, only finitely many of the values in the Zs differ from zero.