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

The quadratic reciprocity theorem for polynomials over a finite field states that $$ \left(\frac{f}{g}\right) \left(\frac{g}{f}\right) = (-1)^{\frac{p-1}{2} \deg(f) \deg(g)}. $$

Bibliography

1

Feng, Ke Qin and Ying, Linsheng, An elementary proof of the law of quadratic reciprocity in $F\sb q(T)$ . 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.