Kronecker symbol KroneckerSymbol 2004-08-24 15:58:02 2004-08-24 16:02:21 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\bf 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 $n$.