Properties of Legendre Symbol

Let $$p$$ be an odd prime.

Let $$a \in \Z$$.

Let $$\left({\frac{a}{p}}\right)$$ be the Legendre symbol: $$\left({\frac{a}{p}}\right) \ \stackrel {\mathbf {def}} {=\!=} \ a^{\left({\frac {p-1}2}\right)} \pmod p$$.

Quadratic Character

 * $$\left({\frac{a}{p}}\right) = 0$$ iff $$a \equiv 0 \pmod p$$;
 * $$\left({\frac{a}{p}}\right) = 1$$ iff $$a$$ is a Quadratic Residue mod $$p$$;
 * $$\left({\frac{a}{p}}\right) = -1$$ iff $$a$$ is a Quadratic Non-Residue mod $$p$$.

Congruent Integers
If $$a \equiv b \pmod p$$, then $$\left({\frac{a}{p}}\right) = \left({\frac{b}{p}}\right)$$.

Multiplicative Nature

 * $$\left({\frac{a b}{p}}\right) = \left({\frac{a}{p}}\right) \left({\frac{b}{p}}\right)$$.

Square is Quadratic Residue

 * $$\left({\frac{a^2}{p}}\right) = 1$$.

Proof of Quadratic Character

 * $$\left({\frac{a}{p}}\right) = 0$$ iff $$a \equiv 0 \pmod p$$;

Follows from Euler's Criterion.


 * $$\left({\frac{a}{p}}\right) = 1$$ iff $$a$$ is a Quadratic Residue mod $$p$$:

This follows directly from the definition of quadratic residue and Euler's Criterion.


 * $$\left({\frac{a}{p}}\right) = -1$$ iff $$a$$ is a Quadratic Non-Residue mod $$p$$:

This follows directly from the definition of Quadratic Non-Residue and Euler's Criterion.

Proof of Congruent Integers
If $$a \equiv b \pmod p$$, then $$\left({\frac{a}{p}}\right) = \left({\frac{b}{p}}\right)$$:

This is just a statement of the quadratic character of congruent integers.

Proof of Multiplicative Nature
$$\left({\frac{a b}{p}}\right) = \left({\frac{a}{p}}\right) \left({\frac{b}{p}}\right)$$:

Follows directly from the identity $$\left({a b}\right)^{\left({\frac {p-1}2}\right)} = a^{\left({\frac {p-1}2}\right)} b^{\left({\frac {p-1}2}\right)}$$.

Proof that Square is Quadratic Residue
$$\left({\frac{a^2}{p}}\right) = 1$$:

Follows directly from the definition.

Alternatively, it also follows from the fact that the Legendre symbol is multiplicative.