An elementary proof of the law of quadratic reciprocity over function fields

Abstract:

Let $P$ and $Q$ be relatively prime monic irreducible polynomials in $\mathbb {F}_{q}[T]$ ($2\nmid q$). In this paper, we give an elementary proof for the following law of quadratic reciprocity in $\mathbb {F}_{q}[T]$: \begin{equation*}\left (\frac {Q}{P}\right )\left (\frac {P}{Q}\right )=(-1)^{\frac {|P|-1}{2}\frac {|Q| -1}{2} },\end{equation*} where $\left (\frac {Q}{P}\right )$ is the Legendre symbol.