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 $n$ .

Title	Kronecker symbol
Canonical name	KroneckerSymbol
Date of creation	2013-03-22 14:33:21
Last modified on	2013-03-22 14:33:21
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	6
Author	mathwizard (128)
Entry type	Definition
Classification	msc 11A07
Classification	msc 11A15
Synonym	Kronecker-Jacobi symbol
Related topic	JacobiSymbol
Related topic	LegendreSymbol