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Legendre symbol (Definition)

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

\begin{displaymath} \left(\frac{a}{p}\right) = \begin{cases} 1 &\text{if }a \tex... ...due }\pmod{p}\ 0 & \text{if } p \text{ divides }a \end{cases}\end{displaymath}

The Legendre symbol can be computed by means of Euler's criterion or Gauss' lemma.

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



"Legendre symbol" is owned by alozano. [ full author list (4) | owner history (3) ]
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See Also: Jacobi symbol, Euler's criterion, quadratic residue, Kronecker symbol, quadratic reciprocity rule, quadratic congruence

Keywords:  Legendre, Character, Jacobi

Attachments:
properties of the Legendre symbol (Theorem) by alozano
values of the Legendre symbol (Theorem) by alozano
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Cross-references: Kronecker symbol, Jacobi symbol, Gauss lemma, Euler's criterion, prime, odd
There are 19 references to this entry.

This is version 9 of Legendre symbol, born on 2001-10-08, modified 2008-01-25.
Object id is 183, canonical name is LegendreSymbol.
Accessed 10051 times total.

Classification:
AMS MSC: 11-00 (Number theory :: General reference works )
Pending Errata and Addenda
None.
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Discussion
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formula by vitriol on 2002-02-16 07:48:05
You might want to add that (a|p) = a^((p-1)/2) (mod p)

however that works in LaTeX :)
[ reply | up ]
hehe latex2html by drini on 2001-10-08 17:25:43
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 
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