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:

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

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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}$

This article, or a section of it, needs explaining. In particular: 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 $q$) or just $\Z$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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.

Also see

• Definition:Character (Number Theory)

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.