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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

$\displaystyle \left({ab\over cd}\right)$ $\textstyle =$ $\displaystyle \left({a\over cd}\right)\left({b\over cd}\right)=\left({ab\over c}\right)\left({ab\over d}\right)$  
  $\textstyle =$ $\displaystyle \left({a\over c}\right)\left({b\over c}\right)\left({a\over d}\right)\left({b\over d}\right)$ (1)

plus additional rules for $m=-1$,
-1 & for $n<0$\cr
1 & for $n>0$,\cr}
\end{displaymath} (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}
\end{displaymath} (3)

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


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.

© 1996-9 Eric W. Weisstein