Jacobi symbol

Noun

Meaning

Jacobi symbol (plural Jacobi symbols)

(number theory) A mathematical function of integer a and odd positive integer b, generally written




(


a
b


)



{\displaystyle \left({a \over b}\right)}

, based on, for each of the prime factors pi of b, whether a is a quadratic residue or nonresidue modulo pi.

Examples

Class number related Peter Gustav Lejeune Dirichlet discovered formulas that relate the class number h of quadratic number fields to the Jacobi symbol. Source: Internet

For example: : Or using a more efficient computation: : The article Jacobi symbol has more examples of Legendre symbol manipulation. Source: Internet

Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. Source: Internet

Related functions * The Jacobi symbol is a generalization of the Legendre symbol that allows for a composite second (bottom) argument n, although n must still be odd and positive. Source: Internet

Then the Jacobi symbol satisfies the law of quadratic reciprocity : : Let λ(n) be Liouville's function. Source: Internet

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