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Tan (The group of rational points on the unit circle, Math. Mag. 69 (1996), 163-171) proved that the group of rational points on the unit circle modulo torsion is isomorphic to finitely infinitely many copies of $\mathbb Z$.
I have given a couple of references to related articles in Kreise und Quadrate modulo $p$, Math. Semesterber. 47 (2000), 51-73.
Tan (The group of rational points on the unit circle, Math. Mag. 69 (1996), 163-171) proved that the group of rational points on the unit circle modulo torsion is isomorphic to finitely many copies of $\mathbb Z$.
I have given a couple of references to related articles in Kreise und Quadrate modulo $p$, Math. Semesterber. 47 (2000), 51-73.