Primitive elements and k-th powers in finite fields

2021/04/25 by Hai-Liang Wu, Wu, Hai-Liang, Yue-Feng She +1
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2104.12185

openalex publication_date 2021/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF_q be the finite field of q elements, and let k| q-1 be a positive integer. Let f(x)=ax²+bx+c be a quadratic polynomial in \mathbbF_q[x] with b²-4ac≠0. In this paper, we show that if q>max\e^e3,(2k)⁶\, then there is a primitive element g of \mathbbF_q such that f(g)∈\mathbbF_q^{× k}=\x^k: x∈\mathbbF_q∖\0\\. Moreover, we shall confirm a conjecture posed by Sun.

Related

• Pair of primitive elements in quadratic form with prescribed trace over a finite field
• On existence of some special pair of primitive elements over finite fields
• Four consecutive primitive elements in a finite field
• Primitive elements of finite fields F_q^r avoiding affine hyperplanes for q=4 and q=5
• Primitive Element Pairs with One Prescribed Trace over a Finite Field