# nonprincipal real characters $\mathrm{@}\text{symoperators}mod\text{tmspace}+.1667em\text{tmspace}+.1667emp$ 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:

${\chi}_{1}$ | ${\chi}_{2}$ | ${\chi}_{3}$ | ${\chi}_{4}$ | |
---|---|---|---|---|

$1$ | $1$ | $1$ | $1$ | $1$ |

$3$ | $1$ | $1$ | $-1$ | $-1$ |

$5$ | $1$ | $-1$ | $1$ | $-1$ |

$7$ | $1$ | $-1$ | $-1$ | $1$ |

Title | nonprincipal real characters $\mathrm{@}\text{symoperators}mod\text{tmspace}+.1667em\text{tmspace}+.1667emp$ are unique |
---|---|

Canonical name | NonprincipalRealCharactersmodPAreUnique |

Date of creation | 2013-03-22 16:34:51 |

Last modified on | 2013-03-22 16:34:51 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 6 |

Author | rm50 (10146) |

Entry type | Theorem |

Classification | msc 11A25 |