Underlying unit group for a Dirichlet character (reviewed)

A Dirichlet character $\chi: \Z\to \C$ has a modulus $n$ such that $\chi$ is induced from a function $\chi:\Z/n\Z\to \C$. It gives a homomorphism $(\Z/n\Z)^\times\to \C^\times$. The underlying unit group is $(\Z/n\Z)^\times$, the group of units of the ring $\Z/n\Z$.