Legendre symbol

Builds on Quadratic residue Prime number

Definition

Let $p$ be an odd prime number and let $a$ be an integer. The Legendre symbol $\left(\frac{a}{p}\right)$ is the number $\left(\frac{a}{p}\right) = \begin{cases} 0 & \text{if } p \mid a, \\ 1 & \text{if } p \nmid a \text{ and } a \text{ is a quadratic residue modulo } p, \\ -1 & \text{if } a \text{ is a quadratic nonresidue modulo } p. \end{cases}$ Here quadratic residue has the meaning of the definition of a quadratic residue.

Remarks

The value depends only on the class of $a$ modulo $p$ . For example, $\left(\frac{2}{7}\right) = 1$ because $3^2 \equiv 2 \pmod 7$ , and $\left(\frac{3}{7}\right) = -1$ . Euler's criterion gives the value as $a^{(p-1)/2}$ modulo $p$ . The law of quadratic reciprocity helps to compute the symbol quickly.

Reading path

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Divisibility definition
Congruence modulo n definition
Prime number definition
Quadratic residue definition
Legendre symbol definition

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17 pages build on this one.

Tag	`01J8AMV9RS`
URI	`m2://01J8AMV9RS`
Kind	definition
Writing	W1 stated
MSC	11A15
Defines	Legendre symbol
Uses	Quadratic residue, Prime number

Legendre symbol

Definition

Remarks

Reading path

Where this leads

Further reading

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