Kronecker Symbol

An extension of the Jacobi Symbol $(n/m)$ to all Integers. It can be computed using the normal rules for the Jacobi Symbol

$$\left({ab\over cd}\right) = \left({a\over cd}\right)\left({b\over cd}\right)=\left({ab\over c}\right)\left({ab\over d}\right)$$

$$= \left({a\over c}\right)\left({b\over c}\right)\left({a\over d}\right)\left({b\over d}\right)$$ (1)

plus additional rules for $m=-1$,

$$(n/-1)=\cases{ -1 & for $n<0$\cr 1 & for $n>0$,\cr}$$ (2)

and $m=2$. The definition for $(n/2)$ is variously written as

$$(n/2) \equiv \cases{ 0 & for $n$\ even\cr 1 & for $n$\ odd... ... $n$\ odd, $n\equiv \pm 3\ \left({{\rm mod\ } {8}}\right)$\cr}$$ (3)

or

$$(n/2)\equiv \cases{ 0 & for $4\vert n$\cr 1 & for $n\equiv... ...({{\rm mod\ } {8}}\right)$\cr {\rm undefined} & otherwise\cr}$$ (4)

(Cohn 1980). Cohn's form ``undefines'' $(n/2)$ for Singly Even Numbers $n\equiv 4\ \left({{\rm mod\ } {2}}\right)$ and $n\equiv -1,3\ \left({{\rm mod\ } {8}}\right)$, probably because no other values are needed in applications of the symbol involving the Discriminants $d$ of Quadratic Fields, where $m>0$ and $d$ always satisfies $d\equiv 0,1\ \left({{\rm mod\ } {4}}\right)$.

The Kronecker Symbol is a Real Character modulo $n$, 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.