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Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $\epsilon$ be a unit of $A$.

My question: Is $\epsilon/\bar{\epsilon}$ a root of unity?

Motivation and Effort This is clear from this question.

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  • $\begingroup$ What's the reason for the downvotes? Unless you tell me, I can't improve my question. $\endgroup$ – Makoto Kato Jul 27 '12 at 7:14
  • $\begingroup$ (I edited your comment into what I think you meant to write. Beware of double negatives!) $\endgroup$ – Willie Wong Jul 27 '12 at 9:02
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Yes. In the notation of this answer, the ratio $\epsilon/\overline{\epsilon}$ lies in $U^-$, which is a finite group (as explained in the linked answer).

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