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
$$\left(\frac{f}{g}\right):=\{\begin{array}{cc}1\hfill & \text{if}f\text{is a square in the quotient ring}F[X]/(g)\text{,}\hfill \\ -1\hfill & \text{otherwise.}\hfill \end{array}$$ |
The quadratic reciprocity theorem for polynomials over a finite field states that
$$\left(\frac{f}{g}\right)\left(\frac{g}{f}\right)={(-1)}^{\frac{p-1}{2}\mathrm{deg}(f)\mathrm{deg}(g)}.$$ |
References
- 1 Feng, Ke Qin and Ying, Linsheng, An elementary proof of the law of quadratic reciprocity in ${F}_{q}\mathit{}\mathrm{(}T\mathrm{)}$. 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 |