Let $F$ be the field ${\mathbb Q}(i)\subset \mathbb C$ and let $T\subset F$ be the set of all elements of complex absolute value 1.
Let $n$ be a natural number $\ge 2$ and let $\mu\_n(T)\subset\mathbb C$ be the set of all $n$-th roots of elements of $T$.
Finally, let $E=F(\mu\_n(T))$. 


Question: Is the field extension $E/F$ finite or infinite?



n-th roots of Pythagorean numbers
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 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.


