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.

Title	quadratic reciprocity for polynomials
Canonical name	QuadraticReciprocityForPolynomials
Date of creation	2013-03-22 12:11:42
Last modified on	2013-03-22 12:11:42
Owner	djao (24)
Last modified by	djao (24)
Numerical id	8
Author	djao (24)
Entry type	Theorem
Classification	msc 11A15
Classification	msc 11T55
Classification	msc 11R58
Related topic	QuadraticReciprocityRule