Definition

Let $b\in\mathbb{Z}$ and $n\in\mathbb{N}$ odd. We factor $n$ as a unique product of prime powers: $$n=p_{1}^{\alpha_{1}}\dots p_{r}^{\alpha_{r}}$$We define the Jacobi Symbol as a product of Legendre symbols via $$\left( \frac{b}{n} \right):=\left( \frac{b}{p_{1}} \right)^{\alpha_{1}}\dots\left( \frac{b}{p_{r}} \right)^{\alpha_{r}}$$