# Kronecker symbol

The 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 $n$.

Title Kronecker symbol KroneckerSymbol 2013-03-22 14:33:21 2013-03-22 14:33:21 mathwizard (128) mathwizard (128) 6 mathwizard (128) Definition msc 11A07 msc 11A15 Kronecker-Jacobi symbol JacobiSymbol LegendreSymbol