Jacobi symbol (reviewed)

The Jacobi symbol \(\displaystyle\left(\frac{a}{b}\right)\) is an extension of the Legendre symbol to all positive odd integers $b$ as follows.

We set $ \displaystyle\left(\frac{a}{1}\right) =1 $ for all $a.$

If $b>1,$ then it has a decomposition into distinct odd primes of the form $b = p_1^{e_1} p_2^{e_2}\cdots p_r^{e_r}$ and the Jacobi symbol is defined as the product of Legendre symbols $$ \displaystyle\left(\frac{a}{b}\right) = \left(\frac{a}{p_1}\right)^{e_1} \left(\frac{a}{p_2}\right)^{e_2}\cdots \left(\frac{a}{p_r}\right)^{e_r}.$$

The Jacobi symbol is multiplicative in the sense that \(\displaystyle\left(\frac{a_1 a_2}{b}\right) = \displaystyle\left(\frac{a_1}{b}\right)\displaystyle\left(\frac{a_2}{b}\right)\) and \(\displaystyle\left(\frac{a}{b_1b_2}\right) = \displaystyle\left(\frac{a}{b_1}\right)\displaystyle\left(\frac{a}{b_2}\right)\) for all integers $a, a_1, a_2$ and all positive odd integers $b, b_1, b_2.$

It has the same special formulas for $a=-1$ and $a = 2$ as the Legendre symbol, namely

$\left(\displaystyle \frac{-1}{b}\right)= (-1)^{\frac{b-1}{2}} = \left\{ \begin{array}{cl} 1 & \text{if } b \equiv 1 \bmod 4 \\ -1 & \text{if } b \equiv -1 \bmod 4 \end{array} \right.$ and $\left(\displaystyle \frac{2}{b}\right) = (-1)^{\frac{b^2-1}{8}}=\left\{ \begin{array}{cl} 1 & \text{if } b \equiv \pm1 \bmod 8 \\ -1 & \text{if } b \equiv \pm3 \bmod 8 \end{array} \right.$.

The law of quadratic reciprocity states that for $m,n$ odd positive coprime integers we have

$$\left(\displaystyle\frac{n}{m}\right) = (-1)^{\frac{m-1}2 \frac{n-1}2} \left(\displaystyle \frac{m}{n}\right).$$