Vipul: Created page with '==Definition== Suppose $p$ is a prime number and $a$ is an integer. The '''Legendre symbol''' of $a$ modulo $p$ , denoted $\l…'$

2010-05-29T22:21:09Z

Created page with '==Definition== Suppose <math>p</math> is a prime number and <math>a</math> is an integer. The '''Legendre symbol''' of <math>a</math> modulo <math>p</math>, denoted <math>\l…'

New page

==Definition==

Suppose <math>p</math> is a [[prime number]] and <math>a</math> is an integer. The '''Legendre symbol''' of <math>a</math> modulo <math>p</math>, denoted <math>\left(\frac{a}{p}\right)</math> is defined to be:

# <math>0</math> if <math>p</math> divides <math>a</math>.
# <math>1</math> if <math>a</math> is a [[defining ingredient::quadratic residue]] modulo <math>p</math>, i.e., <math>p</math> does not divide <math>a</math> and <math>x^2 \equiv a \pmod p</math> has an integer solution for <math>x</math>.
# <math>-1</math> if <math>a</math> is a [[defining ingredient::quadratic nonresidue]] modulo <math>p</math>, i.e., <math>p</math> does not divide <math>a</math> and <math>x^2 \equiv a \pmod p</math> has no integer solution for <math>x</math>.

The Legendre symbol makes sense only relative to a prime number, but it has a generalization called the [[defining ingredient::Jacobi symbol]] which allows the modulus to be any integer. Thus, the Legendre symbol can also be defined as the restriction of the Jacobi symbol to cases where the modulus is prime.

The Legendre symbol is also a special case of a [[Dirichlet character]] modulo <math>p</math>.

==Facts==

* The Legendre symbol is multiplicative in its top portion if the prime is fixed. In other words:

<math>\left(\frac{a_1a_2}{p} \right) = \left(\frac{a_1}{p}\right)\left(\frac{a_2}{p}\right)</math>

* [[Quadratic reciprocity]] relates the Legendre symbol values of two distinct odd primes modulo each other.

Legendre symbol - Revision history

Vipul: Created page with '==Definition== Suppose p is a prime number and a is an integer. The '''Legendre symbol''' of a modulo p, denoted \l…'

Vipul: Created page with '==Definition== Suppose $p$ is a prime number and $a$ is an integer. The '''Legendre symbol''' of $a$ modulo $p$ , denoted $\l…'$