# cases when minus one is a quadratic residue

###### Theorem.

Let $p$ be an odd prime. Then $\mathrm{-}\mathrm{1}$ is a quadratic residue^{} modulo $p$ if and only if $p\mathrm{\equiv}\mathrm{1}\mathrm{mod}\mathrm{4}$.

###### Proof.

Let $p$ be an odd prime. Notice that $p$ is congruent^{} to either $1$ or $3$ modulo $4$. By the definition of the Legendre symbol^{}, we need to verify that
$\left({\displaystyle \frac{-1}{p}}\right)=1$ if and only if $p\equiv 1mod4$. By Euler’s criterion

$$\left(\frac{-1}{p}\right)\equiv {(-1)}^{(p-1)/2}modp.$$ |

Finally, note that the integer $\frac{p-1}{2}$ is even if $p\equiv 1mod4$ and odd if $p\equiv 3mod4$. ∎

Title | cases when minus one is a quadratic residue |
---|---|

Canonical name | CasesWhenMinusOneIsAQuadraticResidue |

Date of creation | 2013-03-22 16:18:10 |

Last modified on | 2013-03-22 16:18:10 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 6 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 11A15 |

Related topic | EulersCriterion |

Related topic | ValuesOfTheLegendreSymbol |