# Jacobi symbol

2010 Mathematics Subject Classification: Primary: 11A15 [MSN][ZBL]

$$\left(\frac aP\right)$$

A function defined for all integers $a$ coprime to a given odd integer $P>1$ as follows: Let $P=p_1\dotsm p_r$ be an expansion of $P$ into prime factors (not necessarily different), then

$$\left(\frac aP\right)=\left(\frac{a}{p_1}\right)\dotsm\left(\frac{a}{p_r}\right),$$

where

$$\left(\frac{a}{p_i}\right)$$

is the Legendre symbol.

The Jacobi symbol is a generalization of the Legendre symbol and has similar properties. In particular, the reciprocity law:

$$\left(\frac PQ\right)\left(\frac QP\right)=(-1)^{(P-1)/2\cdot(Q-1)/2}$$

holds, where $P$ and $Q$ are positive odd coprime numbers, and the supplementary formulas

$$\left(\frac{-1}{P}\right)=(-1)^{(P-1)/2},\quad\left(\frac 2P\right)=(-1)^{(P^2-1)/8}$$

are true.

The Jacobi symbol was introduced by C.G.J. Jacobi (1837).

How to Cite This Entry:
Jacobi symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_symbol&oldid=44571
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article