Primitive values of rational functions at primitive elements of a finite field

2019/09/28 by Stephen D. Cohen, Cohen, Stephen D., Hariom Sharma +3 · 1 citation
Mathematics · #11T23 #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #math.AC #math.NT #msc:11T23

paper · pdf · doi:10.48550/arxiv.1909.13074

12 pages

arxiv created 2019/09/28 · arxiv updated 2019/10/01

Abstract

Given a prime power q and an integer n≥2, we establish a sufficient condition for the existence of a primitive pair (α,f(α)) where α∈ \mathbbF_q and f(x) ∈ \mathbbF_q(x) is a rational function of degree n. (Here f=f₁/f₂, where f₁, f₂ are coprime polynomials of degree n₁,n₂, respectively, and n₁+n₂=n.) For any n, such a pair is guaranteed to exist for sufficiently large q. Indeed, when n=2, such a pair definitely does \em not exist only for 28 values of q and possibly (but unlikely) only for at most 3911 other values of q.

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