Legendre Symbol

definitionnumber-theory

Definition

For integer $a$ and odd prime $p$,

$$\left(\frac{a}{p}\right)=\{\begin{array}{ll}1& \text{ if }a\equiv \square \mathrm{mod}p,a\not\equiv 0\mathrm{mod}p\\ & \\ -1& \text{ if }a\not\equiv \square \mathrm{mod}p,a\not\equiv 0\mathrm{mod}p\\ 0& \text{ if }a\equiv 0\mathrm{mod}p\end{array}$$

Multiplicativity

We have that for any $a,b,p\in \mathbb{Z}$,

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

Examples

See

• Note - Quadratic Reciprocity Law and the Legendre Symbol