Complementaries to Kummer's degree seven reciprocity law and a Dickson Diophantine system

Let Q(ζ) be the cyclotomic field obtained from Q by adjoining a primitive seventh root of unity ζ. Normalized primary elements of this field are characterized and related to Jacobi sums and to solutions of a system of quadratic Diophantine equations of Dickson type involving a rational prime p ≡ 1 (mod 7). These objects and their connection are then used to give another formulation of the complementary laws to Kummer’s reciprocity law of degree seven.