values of the Legendre symbol

For an integer $a$ and an odd prime $p$ , let $\\displaystyle\\left(\\frac{a}{p}\\right)$ be the Legendre symbol.

Theorem.

Let $p$ be an odd prime. The Legendre symbol takes the following values:

1.
$\\left(\\frac{-1}{p}\\right)=\\begin{cases}1&\\text{if }p\\equiv 1\\mod 4\\\\-1&\\text{if }p\\equiv 3\\mod 4.\\end{cases}$
2.
$\\left(\\frac{2}{p}\\right)=\\begin{cases}1&\\text{if }p\\equiv\\pm 1\\mod 8\\\\-1&\\text{if }p\\equiv 3,5\\mod 8.\\end{cases}$
3.
$\\left(\\frac{3}{p}\\right)=\\begin{cases}1&\\text{if }p\\equiv\\pm 1\\mod 12\\\\-1&\\text{otherwise.}\\end{cases}$
4.
$\\left(\\frac{5}{p}\\right)=\\begin{cases}1&\\text{if }p\\equiv\\pm 1\\mod 5\\\\-1&\\text{if }p\\equiv 2,3\\mod 5.\\end{cases}$

Proof.

For a proof of (1), see http://planetmath.org/node/1IsQuadraticResidueIfAndOnlyIfPequiv1Mod4this entry. Part (2) is proved in http://planetmath.org/node/QuadraticCharacterOf2this entry. For parts (3), (4) and (5), we use quadratic reciprocity. For example,

\\left(\\frac{5}{p}\\right)=\\left(\\frac{p}{5}\\right)

and the only quadratic residues modulo $5$ are $\\pm 1\\mod 5$ .∎