quadratic reciprocity for polynomials

Let F be a finite fieldMathworldPlanetmath of characteristicPlanetmathPlanetmath p, and let f and g be distinct monic irreducible (non-constant) polynomialsPlanetmathPlanetmath in the polynomial ring F[X]. Define the Legendre symbolMathworldPlanetmath (fg) by

(fg):={1 if f is a square in the quotient ring F[X]/(g),-1 otherwise.

The quadratic reciprocity theorem for polynomials over a finite field states that



  • 1 Feng, Ke Qin and Ying, Linsheng, An elementary proof of the law of quadratic reciprocity in Fq(T). Sichuan Daxue Xuebao 26 (1989), Special Issue, 36–40.
  • 2 Merrill, Kathy D. and Walling, Lynne H., On quadratic reciprocity over function fieldsMathworldPlanetmath. 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