Power Reciprocity for Binomial Cyclotomic Integers

AbstractWe give an explicit expression for the inversion factor (α/β)l(β/α)−1lof thelth power residue symbol over the cyclotomic field of l th roots of unity, whenαandβare binomial cyclotomic integersx+yζnrelatively prime to each other and tol. Herelis an odd prime number,ζa primitivelth root of unity andx, y∈Z We note that Eisenstein's reciprocity law extends to the case where primary binomial integers replace rational integers. As an application, we obtain necessary and sufficient congruence conditions for a rational integer to be anlth power residue modulo some prime numbers of the form (xl+1)/(x+1)