On primitive roots of unity

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

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On primitive roots of unity

One thought on “On primitive roots of unity

  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.

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