# On primitive roots of unity

Finite subgroups of the group of units of a field are cyclic.

1. By the structure theorem of finitely generated abelian groups, such a subgroup $G$ is isomorphic to $\Bbb{Z}_{d_1}\oplus\dots\oplus\Bbb{Z}_{d_s}$, with $d_1|\dots|d_s$. Then, $x^{d_s}=1$ for all $x\in G$, and since in a field we have at most $d_s$ many $d_s$-th roots of unity, it follows that $|G|\leq d_s$. Therefore $G\simeq \Bbb{Z}_{d_s}$, as we wanted to show.