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 $K$.
My question: Is there a purely imaginary unit in $A$?
EDIT New question: Is the following proposition true? If yes, how would you prove this?
Proposition There is no purely imaginary unit in $A$.
Related questions: