Group of rational points on the unit circle

The set of all 2×2 rotation matrices with rational entries coincides with G. This follows from the fact that the circle group S^1 is isomorphic to \mathrm{SO}(2, \mathbb{R}), and the fact that their rational points coincide. ## Group structure The structure of *G* is an infinite sum of cyclic groups. Let *G*2 denote the subgroup of *G* generated by the point 0 + 1*i*. *G*2 is a cyclic subgroup of order 4. For a prime *p* of form 4*k* + 1, let *G**p* denote the subgroup of elements with denominator *p**n* where *n* is a non-negative integer. *G**p* is an infinite cyclic group, and the point (*a*2 − *b*2)/*p* + (2*ab*/*p*)*i* is a generator of *G**p*. Furthermore, by factoring the denominators of an element of *G*, it can be shown that *G* is a direct sum of *G*2 and the *G**p*. That is: ::G \cong G_2 \oplus \bigoplus_{p \, \equiv \, 1 \, (\text{mod } 4)} G_p. Since it is a direct sum rather than direct product, only finitely many of the values in the *Gp*s are non-zero. ### Example Viewing *G* as an infinite direct sum, consider the element ({*0*}; 2, 0, 1, 0, 0, ..., 0, ...) where the first coordinate *0* is in *C*4 and the other coordinates give the powers of (*a*2 − *b*2)/*p*(*r*) + *i*2*ab*/*p*(*r*), where *p*(*r*) is the *r*th prime number of form 4*k* + 1. Then this corresponds to, in *G*, the rational point (3/5 + *i*4/5)2 ⋅ (8/17 + *i*15/17)1 = −416/425 + i87/425. The denominator 425 is the product of the denominator 5 twice, and the denominator 17 once, and as in the previous example, the square of the numerator −416 plus the square of the numerator 87 is equal to the square of the denominator 425. It should also be noted, as a connection to help retain understanding, that the denominator 5 = *p*(1) is the 1st prime of form 4*k* + 1, and the denominator 17 = *p*(3) is the 3rd prime of form 4*k* + 1. ## The unit hyperbola's group of rational points There is a close connection between this group on the unit hyperbola and the group discussed above. If \frac {a + ib}{c} is a rational point on the unit circle, where *a*/*c* and *b*/*c* are reduced fractions, then (*c*/*a*, *b*/*a*) is a rational point on the unit hyperbola, since (c/a)^2-(b/a)^2=1, satisfying the equation for the unit hyperbola. The group operation here is (x, y) \times (u, v)=(xu+yv, xv+yu),and the group identity is the same point (1, 0) as above. In this group there is a close connection with the hyperbolic cosine and hyperbolic sine, which parallels the connection with cosine and sine in the unit circle group above. ### Copies inside a larger group There are isomorphic copies of both groups, as subgroups (and as geometric objects) of the group of the rational points on the abelian variety in four-dimensional space given by the equation w^2+x^2-y^2+z^2=0. Note that this variety is the set of points with Minkowski metric relative to the origin equal to 0. The identity in this larger group is (1, 0, 1, 0), and the group operation is (a, b, c, d) \times (w, x, y, z)=(aw-bx,ax+bw,cy+dz,cz+dy). For the group on the unit circle, the appropriate subgroup is the subgroup of points of the form (*w*, *x*, 1, 0), with w^2+x^2=1, and its identity element is (1, 0, 1, 0). The unit hyperbola group corresponds to points of form (1, 0, *y*, *z*), with y^2-z^2=1, and the identity is again (1, 0, 1, 0). (Of course, since they are subgroups of the larger group, they both must have the same identity element.) ## References - *The Group of Rational Points on the Unit Circle*[https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1997/0025570x.di021195.02p0087x.pdf](https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1997/0025570x.di021195.02p0087x.pdf), Lin Tan, *Mathematics Magazine* Vol. 69, No. 3 (June, 1996), pp. 163–171 - *The Group of Primitive Pythagorean Triangles*[https://www.jstor.org/pss/2690291](https://www.jstor.org/pss/2690291), Ernest J. Eckert, *Mathematics Magazine* Vol 57 No. 1 (January, 1984), pp 22–26 - ’’Rational Points on Elliptic Curves’’ Joseph Silverman ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Group_of_rational_points_on_the_unit_circle) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Group_of_rational_points_on_the_unit_circle?action=history). ::