PlanetMath: Dirichlet character

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Dirichlet character

(Definition)

A Dirichlet character modulo is a group homomorphism from $\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)^*$ to $\mathbb{C^*}$ . Dirichlet characters are usually denoted by the Greek letter $\chi$ . The function

$\displaystyle \gamma(n)=\begin{cases}\chi(n\bmod m),&\text{if }\gcd(n,m)=1,\\ 0,&\text{if }\gcd(n,m)>1. \end{cases}$

is also referred to as a Dirichlet character. The Dirichlet characters modulo

form a group if one defines $\chi\chi'$ to be the function which takes $a\in\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)^*$ to $\chi(a)\chi'(a)$ . It turns out that this resulting group is isomorphic to $\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)^*$ . The trivial character is given by $\chi(a)=1$ for all $a\in\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)^*$ , and it acts as the identity element for the group. A character $\chi$ modulo

is said to be induced by a character $\chi'$ modulo

if $m'\mid m$ and $\chi(n)=\chi'(n\bmod m')$ . A character which is not induced by any other character is called primitive.

If $\chi$ is non-primitive, the $\gcd$ of all such is called the conductor of $\chi$ .

Examples:

Legendre symbol $\genfrac{(}{)}{}{}{n}{p}$ is a Dirichlet character modulo for any odd prime . More generally, Jacobi symbol $\genfrac{(}{)}{}{}{n}{m}$ is a Dirichlet character modulo .
The character modulo given by $\chi(1)=1$ and $\chi(3)=-1$ is a primitive character modulo . The only other character modulo is the trivial character.

"Dirichlet character" is owned by bbukh. [ full author list (3) | owner history (2) ]

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