Kronecker symbol

The Kronecker symbol is a generalization of the Jacobi symbol to all integers.

Let $n$ be an integer, with prime factorization $u\\cdot{p_{1}}^{e_{1}}\\cdots{p_{k}}^{e_{k}}$ , where $u$ is a unit and the $p_{i}$ are primes. Let $a\\geq 0$ be an integer. The Kronecker symbol $\\left(\\frac{a}{n}\\right)$ is defined to be

\\left(\\frac{a}{n}\\right)=\\left(\\frac{a}{u}\\right)\\prod_{i=1}^{k}\\left(\\frac{a}%{p_{i}}\\right)^{e_{i}}

For odd $p_{i}$ , the number $\\left(\\frac{a}{p_{i}}\\right)$ is simply the usual Legendre symbol. This leaves the case when $p_{i}=2$ . We define $\\left(\\frac{a}{2}\\right)$ by

\\left(\\frac{a}{2}\\right)=\\begin{cases}0&\\text{if $a$ is even}\\\\1&\\text{if $a$ is odd and $n\\equiv 1$ or $n\\equiv 7\\pmod{8}$}\\\\-1&\\text{if $a$ is odd and $n\\equiv 3$ or $n\\equiv 5\\pmod{8}$}\\\\\\end{cases}

Since it extends the Jacobi symbol, the quantity $\\left(\\frac{a}{u}\\right)$ is simply 1 when $u=1$ . When $u=-1$ , we define it by

\\left(\\frac{a}{-1}\\right)=\\begin{cases}-1&\\text{if $a<0$}\\\\1&\\text{if $a>0$}\\\\\\end{cases}

These extensions suffice to define the Kronecker symbol for all integer values $n$ .