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

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 $\mathbb{F}_{q}[T]$:

$$\left (\frac{Q}{P}\right )\left (\frac{P}{Q}\right )=(-1)^{\frac{\vert P\vert-1}{2}\frac{\vert Q\vert -1}{2} },$$

where $\left (\frac{Q}{P}\right )$ is the Legendre symbol.