Kronecker symbol (reviewed)

The Kronecker symbol \(\displaystyle\left(\frac{a}{n}\right)\) is the multiplicative extension of the Jacobi symbol \(\displaystyle\left(\frac{a}{m}\right)\), which is defined on all positive odd integers \(m\), to all integers \(n\in\mathbb{Z}\) as follows.

We set $$\displaystyle\left(\frac{a}{-1}\right) = \left\{ \begin{array}{cl} 1 & \text{if } a\geq 0\\ -1 & \text{if } a < 0; \end{array} \right.$$

$$\displaystyle\left(\frac{a}{2}\right) = \left\{ \begin{array}{cl} 0 & \text{if } a\equiv 0 \bmod 2\\ 1 & \text{if } a\equiv \pm 1 \bmod 8\\ -1 & \text{if } a \equiv \pm 3 \bmod 8; \end{array} \right.$$ and $$\displaystyle\left(\frac{a}{0}\right) = \left\{ \begin{array}{cl} 1 & \text{if } a= \pm 1\\ 0 & \text{otherwise }. \end{array} \right.$$

Then for each nonzero integer $n = \sgn(n) p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r}$ the Kronecker symbol is defined as the product of symbols above and the Legendre symbols for odd primes

$$\displaystyle\left(\frac{a}{n}\right) = \displaystyle\left(\frac{a}{\sgn(n)}\right) \displaystyle\left(\frac{a}{p_1}\right)^{e_1} \displaystyle\left(\frac{a}{p_2}\right)^{e_2}\cdots \displaystyle\left(\frac{a}{p_r}\right)^{e_r}.$$

A Dirichlet character can be written as a Kronecker symbol \(\displaystyle\left(\frac{a}{\cdot}\right)\) if and only if it is real.