quality:corruption, structural:scholar_edition, information:classification, linguistic:paraphrasing, information:filtering
Summary

Let m = p 1 k 1 p 2 k 2 ⋯ = q 1 q 2 ⋯ {\displaystyle m=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots =q_{1}q_{2}\cdots }, p 1 < p 2 < … {\displaystyle p_{1}<p_{2}<\dots } be the factorization of m {\displaystyle m} and assume (r s, m) = 1. {\displaystyle (rs,m)=1.}

There are ϕ (m) {\displaystyle \phi (m)} Dirichlet characters mod m. {\displaystyle m.} They are denoted by χ m, r, {\displaystyle \chi _{m,r},} where χ m, r = χ m, s {\displaystyle \chi _{m,r}=\chi _{m,s}} is equivalent to r ≡ s (mod m). {\displaystyle r\equiv s{\pmod {m}}.} The identity χ m, r (a) χ m, s (a) = χ m, r s (a) {\displaystyle \chi _{m,r}(a)\chi _{m,s}(a)=\chi _{m,rs}(a)\;} is an isomorphism (Z / m Z) × ^ ≅ (Z / m Z) ×. {\displaystyle {\widehat {(\mathbb {Z} /m\mathbb {Z})^{\times }}}\cong (\mathbb {Z} /m\mathbb {Z})^{\times }.}

Each character mod m {\displaystyle m} has a unique factorization as the product of characters mod the prime powers dividing m {\displaystyle m}:

If m = m 1 m 2, (m 1, m 2) = 1 {\displaystyle m=m_{1}m_{2},(m_{1},m_{2})=1} the product χ m 1, r χ m 2, s {\displaystyle \chi _{m_{1},r}\chi _{m_{2},s}} is a character χ m, t {\displaystyle \chi _{m,t}} where t {\displaystyle t} is given by t ≡ r (mod m 1) {\displaystyle t\equiv r{\pmod {m_{1}}}} and t ≡ s (mod m 2). {\displaystyle t\equiv s{\pmod {m_{2}}}.}

Also, χ m, r (s) = χ m, s (r) {\displaystyle \chi _{m,r}(s)=\chi _{m,s}(r)}