\begin{definition}[Definition:Dirichlet Character]
Let $q \in \Z_{>1}$.
Let $\paren {\Z / q \Z}$ denote the ring of integers modulo $q$.
Let $G = \paren {\Z / q \Z}^\times$ be the group of units of $\paren {\Z / q \Z}$.
Let $\C^\times$ be the group of units of $\C$.
A '''Dirichlet character modulo $q$''' is a group homomorphism:
:$\chi: G \to \C^\times$
{{refactor|Presumably the following statement is part of the justification for the generalised definition, which needs to be extracted and put into its own page.|level = medium}}
By Reduced Residue System under Multiplication forms Abelian Group, $a + q \Z \in G$ {{iff}} $\map \gcd {a, q} = 1$.
It is standard practice to extend $\chi$ to a function on $\Z$ by setting:
:$\map \chi A = \begin{cases}
\map \chi {a + q \Z} & : \map \gcd {a, q} = 1 \\
0 & : \text{otherwise}
\end{cases}$
{{explain|The nature of $\chi$ when invoked is not apparent from its notation.  How does one understand, when encountering an instance of $\chi$ whether it means the mapping from $G$ (for some $q$) of just $\Z$?}}
\end{definition}