\begin{definition}[Definition:Jacobi Symbol]
Let $m \in \Z$ be any integer and $n \in \Z$ be any odd integer such that $n \ge 3$.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{i \mathop = 1}^r p_i^{k_i}$.
Then the '''Jacobi symbol''' $\paren {\dfrac m n}$ is defined as:
:$\ds \paren {\frac m n} = \prod_{i \mathop = 1}^r \paren {\frac m {p_i} }^{k_i}$
where $\paren {\dfrac m {p_i} }$ is defined as the Legendre symbol.
\end{definition}