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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 the following proposition true? If yes, how would you prove this?

Proposition There exists a real unit $\eta$ such that $\epsilon = \zeta^g\eta$.

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