Kronecker Symbol

An extension of the Jacobi Symbol to all Integers. It can be computed using the normal rules for the Jacobi Symbol

 (1)

 (2)

and . The definition for is variously written as
 (3)

or
 (4)

(Cohn 1980). Cohn's form undefines'' for Singly Even Numbers and , probably because no other values are needed in applications of the symbol involving the Discriminants of Quadratic Fields, where and always satisfies .

The Kronecker Symbol is a Real Character modulo , and is, in fact, essentially the only type of Real primitive character (Ayoub 1963).

See also Character (Number Theory), Class Number, Dirichlet L-Series, Jacobi Symbol, Legendre Symbol

References

Ayoub, R. G. An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963.

Cohn, H. Advanced Number Theory. New York: Dover, p. 35, 1980.