# nonprincipal real characters $\penalty 0@\mskip 12.0mu {\symoperators mod}\tmspace+{.1667em}\tmspace+{.1667% em}p$ are unique

###### Theorem 1

Let $p$ be a prime. Then there is a unique nonprincipal real Dirichlet character $\chi\mod p$, given by

 $\chi(n)=\left(\frac{n}{p}\right)$

Proof. Note first that $\chi(n)=\left(\frac{n}{p}\right)$ is obviously a nonprincipal real character $\mod p$. Now, suppose $\chi$ is any nonprincipal real character $\mod p$. Choose some generator, $a$, of $(\mathbb{Z}/p\mathbb{Z})^{*}$. Clearly $\chi(a)=-1$ (since otherwise $\chi$ is principal), and thus $\chi(a^{k})=(-1)^{k}$. Since $a^{p-1}=1$, and no lower power of $a$ is $1$, it follows that $\chi$ is $-1$ on exactly $(p-1)/2$ elements of $(\mathbb{Z}/p\mathbb{Z})^{*}$ and is $1$ on exactly $(p-1)/2$ elements. However, since $\chi(x^{2})=\chi(x)^{2}=1$, $\chi$ is $1$ on each of the $(p-1)/2$ squares in $(\mathbb{Z}/p\mathbb{Z})^{*}$. Thus $\chi$ is $1$ on squares and $-1$ on nonsquares, so $\chi(n)=\left(\frac{n}{p}\right)$.

Note that this result is not true if $p$ is not prime. For example, the following is a table of Dirichlet characters modulo $8$, all of which are real:

$\chi_{1}$ $\chi_{2}$ $\chi_{3}$ $\chi_{4}$
$1$ $1$ $1$ $1$ $1$
$3$ $1$ $1$ $-1$ $-1$
$5$ $1$ $-1$ $1$ $-1$
$7$ $1$ $-1$ $-1$ $1$
Title nonprincipal real characters $\penalty 0@\mskip 12.0mu {\symoperators mod}\tmspace+{.1667em}\tmspace+{.1667% em}p$ are unique NonprincipalRealCharactersmodPAreUnique 2013-03-22 16:34:51 2013-03-22 16:34:51 rm50 (10146) rm50 (10146) 6 rm50 (10146) Theorem msc 11A25