# Definition:Jacobi Symbol

## Definition

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:

$\displaystyle n = \prod_{i \mathop = 1}^r p_i^{k_i}$.

Then the Jacobi symbol $\left({\dfrac m n}\right)$ is defined as:

$\displaystyle \left({\frac m n}\right) = \prod_{i \mathop = 1}^r \left({\frac m {p_i}}\right)^{k_i}$

where $\left({\dfrac m {p_i}}\right)$ is defined as the Legendre symbol.

## Notes

It can be seen that the Jacobi symbol is a generalization of the Legendre symbol for a composite denominator.

In order to determine the quadratic character of an integer modulo a composite number, it is necessary to use the expression for the Jacobi symbol as defined above and decompose it into a product of Legendre symbols.

## Source of Name

This entry was named for Carl Gustav Jacob Jacobi.