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

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

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

where $\displaystyle \left({\frac m {p_i}}\right)$ 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.