Let $F$ be a field. If $F^*$ is a cyclic group, must $F$ be a finite field? (It's well-known that if $F$ is a finite field, $F^*$ is a cyclic group).
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2This question has been asked before. I am a bit reluctant to cast my vote for closing this as a duplicate, because it would cause an immediate closure. Summoning an overlord to merge the two, as the other one has no accepted answer. – Jyrki Lahtonen Jun 18 '14 at 10:00