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# Legendre symbol

Legendre Symbol.

Let $p$ be an odd prime. The *Legendre symbol* $\left(\frac{a}{p}\right)$ or $(a|p)$ is defined as:

$\left(\frac{a}{p}\right)=\begin{cases}1&\text{if }a\text{ is a quadratic % residue }\;\;(\mathop{{\rm mod}}p)\\ -1&\text{if }a\text{ is a quadratic nonresidue }\;\;(\mathop{{\rm mod}}p)\\ 0&\text{if }p\text{ divides }a\end{cases}$ |

Generalizations of this symbol are the Jacobi Symbol and the Kronecker symbol.

Keywords:

Legendre, Character, Jacobi

Related:

JacobiSymbol, EulersCriterion, QuadraticResidue, KroneckerSymbol, QuadraticReciprocityRule, QuadraticCongruence

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

11-00*no label found*97U20

*no label found*

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## Comments

## hehe latex2html

did you guys notice that latex2html will render $1$ as a gif, but $0$ as text?

how inconsistent can you be? ;]

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: hehe latex2html

thats really interesting. maybe i'll check out the code to see why it does that. we want to minimize the number of image files embedded. hopefully we will heavily hack latex2html for planetmath, as it does a *lot* we don't need.

-apk

## formula

You might want to add that (a|p) = a^((p-1)/2) (mod p)

however that works in LaTeX :)