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 Φ_p(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.

References

Lang, Serge (1987), Elliptic functions, Graduate Texts in Mathematics, 112 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96508-6, MR 890960

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