# Jacobi symbol

From Encyclopedia of Mathematics

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

where

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).

#### References

[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) |

#### Comments

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]).

#### References

[a1] | D.B. Zagier, "Zetafunktionen und quadratische Körper" , Springer (1981) |

**How to Cite This Entry:**

Jacobi symbol.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Jacobi_symbol&oldid=11678

This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article