In mathematics, Kronecker's congruence, introduced by Kronecker, states that
Φ p ( x , y ) ≡ ( x − y p ) ( x p − y ) mod p , {\displaystyle \Phi _{p}(x,y)\equiv (x-y^{p})(x^{p}-y){\bmod {p}},}
where p is a prime and Φp(x,y) is the modular polynomial of order p, given by
Φ n ( x , j ) = ∏ τ ( x − j ( τ ) ) {\displaystyle \Phi _{n}(x,j)=\prod _{\tau }(x-j(\tau ))}
for j the elliptic modular function and τ running through classes of imaginary quadratic integers of discriminant n.
References Lang, Serge (1987), Elliptic functions, Graduate Texts in Mathematics, vol. 112 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96508-6, MR 0890960