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