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Legendre symbol : ウィキペディア英語版
Legendre symbol



Legendre symbol (''a''/''p'') for various ''a'' (along top) and ''p'' (along left side). Only 0 ≤ ''a'' < ''p'' are shown, since due to the first property below any other ''a'' can be reduced modulo ''p''. Quadratic residues are highlighted in yellow, and correspond precisely to the values 0 and 1.



In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number ''p'': its value on a (nonzero) quadratic residue mod ''p'' is 1 and on a non-quadratic residue (''non-residue'') is −1. Its value on zero is 0.
The Legendre symbol was introduced by Adrien-Marie Legendre in 1798〔A. M. Legendre ''Essai sur la theorie des nombres'' Paris 1798, p 186〕 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol.
== Definition ==
Let ''p'' be an odd prime number. An integer ''a'' is a quadratic residue modulo ''p'' if it is congruent to a perfect square modulo ''p'' and is a quadratic nonresidue modulo ''p'' otherwise. The Legendre symbol is a function of ''a'' and ''p'' defined as
:\left(\frac\right) =
\begin
1 & \text a \text p \text a \not\equiv 0\pmod, \\
-1 & \text a \text p, \\
0 & \text a \equiv 0 \pmod.
\end
Legendre's original definition was by means of the explicit formula
: \left(\frac\right) \equiv a^} \pmod \quad \text \quad\left(\frac\right) \in \.
By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent.〔Hardy & Wright, Thm. 83.〕 Thus Legendre's contribution lay in introducing a convenient ''notation'' that recorded quadratic residuosity of ''a'' mod ''p''. For the sake of comparison, Gauss used the notation a\mathrmp, a\mathrmp according to whether ''a'' is a residue or a non-residue modulo ''p''.
For typographical convenience, the Legendre symbol is sometimes written as (''a''|''p'') or (''a''/''p''). The sequence (''a''|''p'') for ''a'' equal to 0,1,2,... is periodic with period ''p'' and is sometimes called the Legendre sequence, with values occasionally replaced by or .〔Jeong-Heon Kim and Hong-Yeop Song, "Trace Representation of Legendre Sequences," ''Designs, Codes, and Cryptography'' 24, p. 343–348 (2001).〕

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Legendre symbol」の詳細全文を読む



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