  Let $p$ be a prime integer and $n,i$ be positive integers such that
\linebreak $S=\{-1, \ \theta, \ \mu_i=1+\theta+... + \theta^{i-1} \ \mid 1 < i
< \frac{p^n}{2}, \ gcd(p^n,i)=1 \}$ generates the group of units of
$\mathbb{Z}[\theta],$ where $\theta$ is a primitive ${p^n}$--$th$ root of
unity. Denote by $C_{p^n}$ the cyclic group of order $p^n.$ In this paper we
describe explicitly a multiplicatively independent set which generates a
complement to $\pm C_{p^n}$ in the group of units of the integral group ring of
$C_{p^n}.$
