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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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    (I edited your comment into what I think you meant to write. Beware of [double negatives](http://en.wikipedia.org/wiki/Double_negative)!)2012-07-27

2 Answers 2

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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).