Group of rational points on the unit circle

The group of rational points on the unit circle consists of the points with rational coordinates for which applies. The set of these points is closely related to the prime Pythagorean triples . If a primitive right-angled triangle with integer coprime side lengths is given, where is the hypotenuse , then there is the rational point on the unit circle . Conversely, if a rational point is on the unit circle, then there is a primitive right-angled triangle with the sides , the least common multiple being the denominator of and . ${\displaystyle C(\mathbb {Q} )}$ ${\displaystyle (x,y)}$ ${\displaystyle x^{2}+y^{2}=1}$ ${\displaystyle a,b,c}$ ${\displaystyle c}$ ${\displaystyle ({\tfrac {a}{c}},{\tfrac {b}{c}})}$ ${\displaystyle (x,y)}$ ${\displaystyle xc,yc,c}$ ${\displaystyle c}$ ${\displaystyle x}$ ${\displaystyle y}$

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 angle addition if and , where the angle of the radius vector with the radius vector is in the mathematically positive sense. So if and in each case with the angles and form, their sum is the rational point on the unit circle with the angle in the sense of the usual addition of angles. ${\displaystyle (1,0)}$ ${\displaystyle (x,y)+(t,u)=(xt-uy,xu+yt)}$ ${\displaystyle x=\cos(\alpha )}$ ${\displaystyle y=\sin(\alpha )}$ ${\displaystyle \alpha }$ ${\displaystyle (x,y)}$ ${\displaystyle (1,0)}$ ${\displaystyle (x,y)}$ ${\displaystyle (t,u)}$ ${\displaystyle (1,0)}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle (xt-uy,xu+yt)}$ ${\displaystyle \alpha +\beta }$

If you identify the point with the complex number , the addition in the multiplication corresponds to in . ${\displaystyle (x,y)}$ ${\displaystyle x+yi}$ ${\displaystyle C(\mathbb {Q} )}$ ${\displaystyle \mathbb {C} }$

Group structure

The group is isomorphic to an infinite direct sum of cyclic subsets of : ${\displaystyle C(\mathbb {Q} )}$ ${\displaystyle C(\mathbb {Q} )}$

${\displaystyle C(\mathbb {Q} )\cong C_{2}\oplus \left(\bigoplus _{p\in \mathbb {P} , \atop p\equiv 1{\pmod {4}}}C_{p}\right),}$

where is the subgroup generated by , and which are those subgroups which are co- generated by points of the form , where is a Pythagorean prime number . ${\displaystyle C_{2}}$ ${\displaystyle (0,1)}$ ${\displaystyle C_{p}}$ ${\displaystyle \left({\tfrac {a^{2}-b^{2}}{p}},{\tfrac {2ab}{p}}\right)}$ ${\displaystyle a,b\in \mathbb {N} ,a>b>0,a^{2}+b^{2}=p}$ ${\displaystyle p}$

This statement is an application of Hilbert's Theorem 90 to the problem of the rational points on the unit circle, see also: Lin Tan.

literature

• Lin Tan: The Group of Rational Points on the Unit Circle. In: Mathematics Magazine. Vol. 69, No. 3, June 1996, pp. 163–171, doi : 10.2307 / 2691462 , digitized version (PDF; 792 kB) ( memento of March 8, 2012 in the Internet Archive ) or directly https: //www.maa .org / sites / default / files / pdf / upload_library / 22 / Allendoerfer / 1997 / 0025570x.di021195.02p0087x.pdf .
• Ernest J. Eckert: The Group of Primitive Pythagorean Triangles. In: Mathematics Magazine. Vol. 57, No. 1, January 1984, pp. 22-26, doi : 10.2307 / 2690291 .