nLab Jacobi symbol

Idea

For an odd integer $n$, and $a$ relatively prime to $n$, the Jacobi symbol $\left(\frac{a}{n}\right)$ (notated below as $(a \vert n)$) is the sign of the permutation on the group of units in the integers modulo $n$ where the permutation is given by multiplying by $a$.

If $a$ and $n$ share a prime factor, then $(a \vert n)$ is set to $0$ by default.

Properties

• If $n = p$ is an odd prime, then the value of $(a \vert p)$ is the same as for the Legendre symbol $\left(\frac{a}{p}\right)$.

• If the prime factorization of an odd integer $n$ is $p_1 p_2 \ldots p_k$, then

  $$(a \vert n) = \left(\frac{a}{p_1}\right)\left(\frac{a}{p_2}\right)\cdots \left(\frac{a}{p_k}\right).$$

• The quadratic reciprocity law holds in the generality of the Jacobi symbol: if $m, n$ are relatively prime odd integers, then

  $$(m \vert n)(n \vert m) = (-1)^{\frac{m-1}{2} \cdot \frac{n-1}{2}}.$$

Related concepts

• quadratic reciprocity law

References

See also:

• Wikipedia: Jacobi symbol