quadratic reciprocity for polynomials

Let $F$ be a finite field of characteristic $p$ , and let $f$ and $g$ be distinct monic irreducible (non-constant) polynomials in the polynomial ring $F[X]$ . Define the Legendre symbol $\\left(\\frac{f}{g}\\right)$ by

\\left(\\frac{f}{g}\\right):=\\begin{cases}1&\\text{ if $f$ is a square in the %quotient ring $F[X]/(g)$,}\\\\-1&\\text{ otherwise.}\\end{cases}

The quadratic reciprocity theorem for polynomials over a finite field states that

\\left(\\frac{f}{g}\\right)\\left(\\frac{g}{f}\\right)=(-1)^{\\frac{p-1}{2}\\deg(f)%\\deg(g)}.

References

1 Feng, Ke Qin and Ying, Linsheng,An elementary proof of the law of quadratic reciprocity in $F_{q}(T)$ .Sichuan Daxue Xuebao 26 (1989), Special Issue, 36–40.
2 Merrill, Kathy D. and Walling, Lynne H.,On quadratic reciprocity over function fields.Pacific J. Math. 173 (1996), no. 1, 147–150.