Definition:Dirichlet Character

Definition
Let $G = \left( \Z / q\Z \right)^\times$ be the the multiplicative Group of Units of a the ring ring $\left( \Z / q\Z \right)$.

A Dirichlet character modulo q is a  group homomorphism


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

By Multiplicative Group of Integers Modulo m, $a + q \Z \in G$ if and only if $\gcd(a,q) = 1$.

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


 * $\displaystyle

\chi(a) = \left\{ \begin{array}{rl} \chi(a + q \Z),& \gcd(a,q) = 1\\ 0,&\text{otherwise} \end{array} \right. $

Trivial Characters
If $\chi(a) = 1$ for all $a \in G$, then we call $\chi$ the trivial or principal character modulo $q$.

Odd Characters
If $\chi(-1) = -1$ then we call $\chi$ odd, otherwise $\chi$ is even.

Primitive Characters
Let $q^*$ be the least divisor of $q$ such that we can write $\chi = \chi_0 \chi^*$, where $\chi_0$ is the trivial character modulo $q$, and $\chi^*$ is some character modulo $q^*$.

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

Also See

 * Character (Number Theory)