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

In 1991, Rousseau gave a new proof of Gauss’s quadratic reciprocity by comparing two distinct coset representations of the group $({\mathbb{Z}}_p^{\ast }\times {\mathbb{Z}}_q^{\ast })\mathrm{/}U$ 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 $q-1$.