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 $\pm 1\mod 5$ . ∎

Title	values of the Legendre symbol
Canonical name	ValuesOfTheLegendreSymbol
Date of creation	2013-03-22 16:18:13
Last modified on	2013-03-22 16:18:13
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Theorem
Classification	msc 11-00
Related topic	1IsQuadraticResidueIfAndOnlyIfPequiv1Mod4
Related topic	QuadraticCharacterOf2