Definition:Character (Number Theory)

Definition
Let $G$ be a finite abelian group.

A character of $G$ is a group homomorphism


 * $\chi : G \to C^\times$

where $\C^\times$ is the multiplicative Group of Units of $\C$.

The character


 * $\chi_0 : g \mapsto 1,\quad \forall g \in G$

is called the trivial character.

Note
In algebra, character refers to the trace of a representation of $G$. This generalizes the number theorist's definition above.