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

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