An elementary proof of the d-th power reciprocity law over function fields

Abstract

This paper generalises the proof of quadratic reciprocity law in Fq[T] presented by C. G. Ji and Y. Xue [Proc. Amer. Math. Soc. 136 (2008), no. 9, 3035–3039; MR2407064] to the case of d-th power residues, where d divides the order of F∗q. Using only elementary properties of finite fields and basic number-theoretic tools we show that if P,Q∈Fq[T] are distinct irreducible polynomials then (PQ)d=(−1)q−1ddeg(P)deg(Q)(QP)d, where (PQ)d is the d-th power residue symbol