# Definition:Dirichlet Character

## Definition

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:

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

By Reduced Residue System under Multiplication forms Abelian Group, $a + q \Z \in G$ if and only if $\map \gcd {a, q} = 1$.

It is standard practice to extend $\chi$ to a function on $\Z$ by setting:

$\map \chi A = \begin{cases} \map \chi {a + q \Z} & : \map \gcd {a, q} = 1 \\ 0 & : \text{otherwise} \end{cases}$

### Primitive Character

Let $\chi_0$ be the trivial (Dirichlet) character modulo $q$.

Let $q^*$ be the least divisor of $q$ such that:

$\chi = \chi_0 \chi^*$

where $\chi^*$ is some character modulo $q^*$.

If $q = q^*$ then $\chi$ is called primitive, otherwise $\chi$ is imprimitive.

## Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.