An elementary proof of the d-th power reciprocity law over function fields - Blaszczok, Anna (Annales Mathematicae Silesianae, 2011)
This paper generalises the proof of quadratic reciprocity law in ????q[T] presented by Chun-Gang Ji and Yan Xue [2] to the case of d-th power residues, where d divides the order of ????*q. Using only elementary properties of finite fields and basic number-theoretic tools we show that if P,Q∈????q[T] are distinct irreducible polynomials then (P/Q)d = (-1)q-1/ddeg(P)deg(Q)(Q/P)d, where (P/Q)d is the d-th power residue symbol.