Classical Congruences for Parameters in Binary Quadratic Forms

Let ( ) be an imaginary quadratic field with discriminant −k and class number h, with k≠3, 4, or 8. Let p be a prime such that ( )=1. There are integers C, D, unique up to sign, such that 4ph=C2+kD2, p C. Stickelberger gave a congruence for C modulo p which extends congruences of Gauss, Jacobi, and Eisenstein. Stickelberger also gave a simultaneous congruence for C modulo k, but only for prime k. We prove an extension of his result that holds for all k, giving along the way an exposition of his work.