structural:reformatting
Elementary facts

4) Since gcd (1, m) = 1, {\displaystyle \gcd(1,m)=1,} property 2) says χ (1) ≠ 0 {\displaystyle \chi (1)\neq 0} so it can be canceled from both sides of χ (1) χ (1) = χ (1 × 1) = χ (1) {\displaystyle \chi (1)\chi (1)=\chi (1\times 1)=\chi (1)}:

5) Property 3) is equivalent to

6) Property 1) implies that, for any positive integer n {\displaystyle n}

7) Euler's theorem states that if (a, m) = 1 {\displaystyle (a,m)=1} then a ϕ (m) ≡ 1 (mod m). {\displaystyle a^{\phi (m)}\equiv 1{\pmod {m}}.} Therefore,

That is, the nonzero values of χ (a) {\displaystyle \chi (a)} are ϕ (m) {\displaystyle \phi (m)} -th roots of unity:

for some integer r {\displaystyle r} which depends on χ, ζ, {\displaystyle \chi,\zeta,} and a {\displaystyle a}. This implies there are only a finite number of characters for a given modulus.

8) If χ {\displaystyle \chi } and χ ′ {\displaystyle \chi '} are two characters for the same modulus so is their product χ χ ′, {\displaystyle \chi \chi ',} defined by pointwise multiplication:

The principal character is an identity:

9) Let a − 1 {\displaystyle a^{-1}} denote the inverse of a {\displaystyle a} in (Z / m Z) × {\displaystyle (\mathbb {Z} /m\mathbb {Z})^{\times }}.Then

The complex conjugate of a root of unity is also its inverse (see here for details), so for (a, m) = 1 {\displaystyle (a,m)=1}

Thus for all integers a {\displaystyle a}

10) The multiplication and identity defined in 8) and the inversion defined in 9) turn the set of Dirichlet characters for a given modulus into a finite abelian group.