Abstract :
We 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, yset membership, variantimage We note that Eisensteinʹs 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).
