Units of $\mathbb{Z}{C}_{p^n}$

Abstract

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 $C_{p^n}.$