Editing Dirichlet character (section)

== Elementary facts ==

4) Since  <math>\gcd(1,m)=1,</math> property 2) says <math>\chi(1)\ne 0</math> so it can be canceled from both sides of  <math>\chi(1)\chi(1)=\chi(1\times 1) =\chi(1)</math>:

:<math>\chi(1)=1.</math><ref>These properties are derived in all introductions to the subject, e.g. Davenport p. 27, Landau p. 109.</ref>

5) Property 3) is equivalent to

:if <math>a \equiv b \pmod{m}</math> &nbsp; then <math>\chi(a) =\chi(b).</math>

6) Property 1) implies that, for any positive integer <math>n</math>
:<math>\chi(a^n)=\chi(a)^n.</math>

7) [[Euler's theorem]] states that if <math>(a,m)=1</math> then <math>a^{\phi(m)}\equiv 1 \pmod{m}.</math> Therefore,
:<math>\chi(a)^{\phi(m)}=\chi(a^{\phi(m)})=\chi(1)=1.</math>

That is, the nonzero values of <math>\chi(a)</math> are <math>\phi(m)</math>-th [[root of unity|roots of unity]]:

:<math>
\chi(a)=
\begin{cases}
0 &\text{if } \gcd(a,m)>1\\
\zeta_{\phi(m)}^r&\text{if } \gcd(a,m)=1
\end{cases}</math>

for some integer <math>r</math> which depends on <math>\chi, \zeta,</math> and <math>a</math>. This implies there are only a finite number of characters for a given modulus.

8) If <math>\chi</math> and <math>\chi'</math> are two characters for the same modulus so is their product <math>\chi\chi',</math> defined by pointwise multiplication:
:<math>\chi\chi'(a) = \chi(a)\chi'(a)</math> &nbsp; (<math>\chi\chi'</math> obviously satisfies 1-3).<ref>In general, the product of a character mod <math>m</math> and a character mod <math>n</math> is a character mod <math>\operatorname{lcm}(m,n)</math></ref>

The principal character is an identity:
:<math>
\chi\chi_0(a)=\chi(a)\chi_0(a)=
\begin{cases}
0 \times 0 &=\chi(a)&\text{if } \gcd(a,m)>1\\
\chi(a)\times 1&=\chi(a) &\text{if } \gcd(a,m)=1.
\end{cases}</math>

9) Let <math>a^{-1}</math> denote the inverse of <math>a</math>  in <math>(\mathbb{Z}/m\mathbb{Z})^\times</math>.
Then
:<math>\chi(a)\chi(a^{-1})=\chi(aa^{-1})=\chi(1)=1,
</math> so <math>\chi(a^{-1})=\chi(a)^{-1},</math> which extends 6) to all integers.

The [[complex conjugate]] of a root of unity is also its inverse (see [[root of unity#Elementary properties|here]] for details), so for <math>(a,m)=1</math>
:<math>\overline{\chi}(a)=\chi(a)^{-1}=\chi(a^{-1}). </math> &nbsp; (<math>\overline\chi</math> also obviously satisfies 1-3).

Thus for all integers <math>a</math>
:<math>
\chi(a)\overline{\chi}(a)=
\begin{cases}
0 &\text{if } \gcd(a,m)>1\\
1 &\text{if } \gcd(a,m)=1
\end{cases}; </math> &nbsp; in other words  <math>\chi\overline{\chi}=\chi_0</math>.&nbsp;

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 [[abelian group#Finite abelian groups|finite abelian group]].