An elementary proof of the d-th power reciprocity law over function fields
Abstract
This paper generalises the proof of quadratic reciprocity law in 𝔽q[T] presented by Chun-Gang Ji and Yan Xue [2] to the case of d-th power residues, where d divides the order of 𝔽*q. Using only elementary properties of finite fields and basic number-theoretic tools we show that if P,Q∈𝔽q[T] are distinct irreducible polynomials then
(P/Q)d = (-1)q-1/ddeg(P)deg(Q)(Q/P)d,
where (P/Q)d is the d-th power residue symbol.
Keywords
polynomial ring; d-th power residue; reciprocity law
References
2. Chun-Gang J., Yan X., An elementary proof of the law of quadratic reciprocity over function fields, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3035–3039.
3. Dedekind R., Abriss einer Theorie der höheren Congruenzen in Bezug auf einer rellen Primzahl-Modulus, J. Reine Angew. Math. 54 (1857), 1–26.
4. Lidl R., Niederreiter H., Finite fields, Cambridge University Press, Cambridge, 2008.
5. Rosen M., Number theory in function fields, Springer-Verlag, New York, 2002.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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