Definition:Dirichlet Character/Primitive Character
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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 $\C$.
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.