Show that for any $z \in \mathbb S^1$ the function $\chi_z$ defined by $n \mapsto z^n$ is a character of $\mathbb Z.$
I am familiar with character of a group representation. Here in order to prove that $\chi_z$ is a character all we need to find out is the underlying group representation of $\mathbb Z$ which is nothing but a group homomorphism $\rho : \mathbb Z \longrightarrow GL(V),$ for some vector space $V$ over some field $\mathbb F.$ I think that if we take the one dimensional representation of $\mathbb Z$ given by $n \mapsto z^n$ then we are through. Is it what is needed to be shown? But I don't understand what's the role of $\mathbb S^1$ here. It seems to me that for any $z \in \mathbb C \setminus \{0\}$ the same reasoning would work. Could anyone please check my argument above and confirm whether it is fine or not?
Thanks a bunch!