I am reading algebraic number theory and got stuck by this little problem. Let $\zeta$ be the $p$-th root of unity, where $p$ is a prime number. Let $r,s$ be two integers that are coprime to $p$. Prove that $\frac{\zeta^r-1}{\zeta^s-1}\in O_F^{\times}$, where $O_F$ is the ring of algebraic integers in $F = \mathbb{Q}(\zeta)$.
How do I construct a polynomial with integer coefficients that has this fraction as one of its roots? Moreover, this fraction has to be a unit in $O_F$.
