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

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# On primitive roots of unity

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

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By the structure theorem of finitely generated abelian groups, such a subgroup is isomorphic to , with . Then, for all , and since in a field we have at most many -th roots of unity, it follows that . Therefore , as we wanted to show.

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