A reciprocity theorem for the case n = 3 solved by Gauss using "integers" of the form a + b ρ, when ρ is a root of x^2 + x + 1 = 0 (i.e., ρ equals -(-1)^(1/3) or (-1)^(2/3)) and a, b are integers.

cubic residue | reciprocity theorem