# 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 $\left(\frac{f}{g}\right)$ 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., Pacific J. Math. 173 (1996), no. 1, 147–150.
Title quadratic reciprocity for polynomials QuadraticReciprocityForPolynomials 2013-03-22 12:11:42 2013-03-22 12:11:42 djao (24) djao (24) 8 djao (24) Theorem msc 11A15 msc 11T55 msc 11R58 QuadraticReciprocityRule