cyclotomic units

Let $L=\mathbb{Q}({\zeta }_m)$ be a cyclotomic extension of $\mathbb{Q}$ with $m$ chosen to be minimal. Then the ring of integers is given by $\mathbb{Z}({\zeta }_m)$, and we denote the group of units by ${𝒪}_L^{\times }$. The cyclotomic units are the elements of a subgroup $C$ of ${𝒪}_L^{\times }$ given by

$\eta ={\displaystyle \frac{{\zeta }_m^r-1}{{\zeta }_m^s-1}}$

with $r$ and $s$ relatively prime to $m$.