I am trying to prove the following:
Let $l$ be a prime and let $\zeta$ be a $l$th root of unity. Show that, in the order $\{ 1, \zeta, \ldots, \zeta^{l-2} \}$ of the field $\mathbb{Q}(\zeta)$, if a product $\alpha \beta$ is divisible by $1-\zeta$, then $\alpha$ or $\beta$ must be divisible by $1-\zeta$.
I know that $1-\zeta$ is irreducible in the maximal order $\mathbb{Z}[ \zeta ]$, and I am trying to mimic the proof of that statement, but I'm stuck.
Can someone please help?
I see that all the products I am dealing with are of the form $\zeta^k$.
Also does the order contain $\zeta^{l-1}$?
I have a feeling that "the order ${ 1, \zeta, \ldots, \zeta^{l-2} }$" is actually just $\mathbb{Z}[ \zeta ]$. Is this true?