Jacobi symbol

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A function defined for all integers coprime to a given odd integer as follows: Let be an expansion of into prime factors (not necessarily different), then


is the Legendre symbol.

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

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

are true.

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


[1] C.G.J. Jacobi, "Gesammelte Werke" , 1–7 , Reimer (1881–1891)
[2] P.G.L. Dirichlet, "Vorlesungen über Zahlentheorie" , Vieweg (1894)
[3] P. Bachmann, "Niedere Zahlentheorie" , 1–2 , Teubner (1902–1910)


Considered as a function on , the Jacobi symbol is an example of a real character. This real character plays an important role in the decomposition of rational primes in a quadratic field (see [a1]).


[a1] D.B. Zagier, "Zetafunktionen und quadratische Körper" , Springer (1981)
How to Cite This Entry:
Jacobi symbol. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article