Kronecker's congruence

In mathematics, Kronecker's congruence, introduced by Kronecker, states that
 * $$ \Phi_p(x,y)\equiv (x-y^p)(x^p-y)\bmod p,$$

where p is a prime and Φp(x,y) is the modular polynomial of order p, given by
 * $$\Phi_n(x,j) = \prod_\tau (x-j(\tau))$$

for j the elliptic modular function and τ running through classes of imaginary quadratic integers of discriminant n.