Reciprocity by resultant in $k[t]$

Abstract

Let $k$ be a perfect field with procyclic absolute Galois group and containing a primitive $n$-th root of unity. We define a degree $n$ power residue symbol $(\frac{a}{b}{)}_n$in the ring $k[t]$, show that it is equal to “the character of the resultant Res$b,a$” and deduce a reciprocity law. We are motivated by commonalities between the classical case $k={\mathbb{F}}_q$ and the novel but very simple case $k=\mathbb{R}$.