Definition:Legendre Symbol/Definition 2
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Definition
Let $p$ be an odd prime.
Let $a \in \Z$.
The Legendre symbol $\paren {\dfrac a p}$ is defined as:
\(\ds 0 \) | if $a \equiv 0 \pmod p$ | ||||||||
\(\ds +1 \) | if $a$ is a quadratic residue of $p$ | ||||||||
\(\ds -1 \) | if $a$ is a quadratic non-residue of $p$ |
Also see
- Results about the Legendre symbol can be found here.
Source of Name
This entry was named for Adrien-Marie Legendre.
Historical Note
The Legendre symbol was introduced by Adrien-Marie Legendre in Paris in $1798$, during his partly successful attempt to prove the Law of Quadratic Reciprocity.
The function was later expanded into the Jacobi symbol, the Kronecker symbol, the Hilbert symbol and the Artin symbol.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Legendre symbol
- 2005: Richard Crandall and Carl B. Pomerance: Prime Numbers: A Computational Perspective (2nd ed.): Definition $2.3.2$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Legendre symbol