Hasse's Resolution Modulus Theorem

The Jacobi Symbol $(a/y)=\chi(y)$ as a Character can be extended to the Kronecker Symbol $(f(a)/y)=\chi^*(y)$ so that $\chi^*(y)=\chi(y)$ whenever $\chi(y)\not=0$. When $y$ is Relatively Prime to $f(a)$, then $\chi^*(y)\not=0$, and for Nonzero values $\chi^*(y_1)=\chi^*(y_2)$ Iff $y_1\equiv y_2{\rm\ mod}^+\ f(a)$. In addition, $\vert f(a)\vert$ is the minimum value for which the latter congruence property holds in any extension symbol for $\chi(y)$.

See also Character (Number Theory), Jacobi Symbol, Kronecker Symbol

References

Cohn, H. Advanced Number Theory. New York: Dover, pp. 35-36, 1980.