nonprincipal real characters $\mathrm{@}mod+.1667em+.1667emp$ are unique

Theorem 1

Let $p$ be a prime. Then there is a unique nonprincipal real Dirichlet character $\chi \mathrm{mod}p$, 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 $modp$. Now, suppose $\chi$ is any nonprincipal real character $modp$. 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:

	${\chi }_1$	${\chi }_2$	${\chi }_3$	${\chi }_4$
$1$	$1$	$1$	$1$	$1$
$3$	$1$	$1$	$-1$	$-1$
$5$	$1$	$-1$	$1$	$-1$
$7$	$1$	$-1$	$-1$	$1$