| Version 8 |
Version 7 |
| \textbf{Legendre Symbol.}\\ |
\textbf{Legendre Symbol.}\\ |
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Let $p$ be an odd prime. The \emph{Legendre symbol} $\left(\frac{a}{p}\right)$ or $(a|p)$ is defined as:
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Let $p$ be an odd prime. The \emph{Legendre symbol} $\left(\frac{a}{p}\right)$ or $(a\mid p)$ is defined as:
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| \[ |
\[ |
| \left(\frac{a}{p}\right) = |
\left(\frac{a}{p}\right) = |
| \begin{cases} |
\begin{cases} |
| 1 &\text{if }a \text{ is a quadratic residue }\pmod{p}\\ |
1 &\text{if }a \text{ is a quadratic residue }\pmod{p}\\ |
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-1 &\text{if }a \text{ is a quadratic nonresidue }\pmod{p}\\
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-1 &\text{if }a \text{ is a non-quadratic residue }\pmod{p}\\
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| 0 & \text{if } p \text{ divides }a |
0 & \text{if } p \text{ divides }a |
| \end{cases} |
\end{cases} |
| \] |
\] |
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| The Legendre symbol can be computed by means of Euler's criterion or Gauss' lemma. |
The Legendre symbol can be computed by means of Euler's criterion or Gauss' lemma. |
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Generalizations of this symbol are the Jacobi Symbol and the Kronecker symbol.
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A generalization of this symbol is the Jacobi Symbol.
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