|
|
|
|
quadratic reciprocity for polynomials
|
(Theorem)
|
|
|
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 $$ \left(\frac{f}{g}\right) \left(\frac{g}{f}\right) = (-1)^{\frac{p-1}{2} \deg(f) \deg(g)}. $$
- 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.
|
"quadratic reciprocity for polynomials" is owned by djao.
|
|
(view preamble | get metadata)
Cross-references: theorem, quadratic reciprocity, Legendre symbol, polynomial ring, polynomials, irreducible, monic, characteristic, finite field
This is version 4 of quadratic reciprocity for polynomials, born on 2002-01-17, modified 2003-09-15.
Object id is 1488, canonical name is QuadraticReciprocityForPolynomials.
Accessed 3301 times total.
Classification:
| AMS MSC: | 11A15 (Number theory :: Elementary number theory :: Power residues, reciprocity) | | | 11T55 (Number theory :: Finite fields and commutative rings :: Arithmetic theory of polynomial rings over finite fields) | | | 11R58 (Number theory :: Algebraic number theory: global fields :: Arithmetic theory of algebraic function fields) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![\begin{displaymath} \left(\frac{f}{g}\right) := \begin{cases} 1 & \text{ if $f$ ... ...tient ring $F[X]/(g)$,} \ -1 & \text{ otherwise.} \end{cases}\end{displaymath}](http://images.planetmath.org:8080/cache/objects/1488/js/img1.png "\begin{displaymath} \left(\frac{f}{g}\right) := \begin{cases} 1 & \text{ if $f$ ... ...tient ring $F[X]/(g)$,} \ -1 & \text{ otherwise.} \end{cases}\end{displaymath}")