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:

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

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.

Also see

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.

Sources

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