On the reciprocity law in $\mathbb{F}_{q}[t]$

TL;DR

This paper extends Rousseau's elementary proof of quadratic reciprocity to the polynomial ring over finite fields, establishing a general reciprocity law for the $d$th power residue symbol.

Contribution

It generalizes Rousseau's approach to $F_q[t]$, providing a new proof for the reciprocity law applicable to any divisor of $q-1$.

Findings

Extended Rousseau's method to $F_q[t]$

Proved reciprocity law for $d$th power residues

Provided an elementary proof without Gauss's Lemma

Abstract

In 1991, Rousseau gave a new proof of Gauss's quadratic reciprocity by comparing two distinct coset representations of the group $(\mathbb{Z}_{p}^{*} \times \mathbb{Z}_{q}^{*}) / U$ using the Chinese Remainder Theorem, without Gauss's Lemma. In this paper, we extend Rousseau's approach to $\mathbb{F}_{q}[t]$, providing a new, elementary proof of the reciprocity law for the $d$th power residue symbol, where $d$ is any divisor of $q-1$.