8. Show that if $a$ and $b$ are odd integers and $(a, b)=1$, then the following reciprocity law holds for the Jacobi symbol:
$$\left(\frac{a}{|b|}\right)\left(\frac{b}{|a|}\right)=\left\{\begin{array}{ll} -(-1)^{\frac{a-1}{2} \frac{b-1}{2}} & \text { if } a<0 \text { and } b<0 \\ (-1)^{\frac{a-1}{2} \frac{b-1}{2}} & \text { otherwise. } \end{array}\right.$$

In problems $9-15$ we deal with the Kronecker symbol which is defined as follows. Let $a$ be a positive integer that is not a perfect square such that $a \equiv 0$ or $1(\bmod 4)$. We define
$$\begin{array}{l} \left(\frac{a}{2}\right)=\left\{\begin{aligned} 1 & \text { if } a \equiv 1(\bmod 8) \\ -1 & \text { if } a \equiv 5(\bmod 8) . \end{aligned}\right. \\ \left(\frac{a}{p}\right)=\text { the Legendre symbol }\left(\frac{a}{p}\right) \text { if } p \text { is an odd prime such that } p \nmid a \text {. } \\ \left(\frac{a}{n}\right)=\prod_{j=1}^{r}\left(\frac{a}{p_{j}}\right)^{t,} \text { if }(a, n)=1 \text { and } n=\prod_{j=1}^{r} p_{j}^{t} \text { is the prime factorization of } n \text {. } \end{array}$$