The modified prime sieve for primitive elements in finite fields

2025/07/29 by Bagger, Gustav Kjærbye, Punch, James · 1 citation
#11T24 #12E20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.21515

Abstract

Let r ≥ 2 be an integer, q a prime power and \mathbbF_q the finite field with q elements. Consider the problem of showing existence of primitive elements in a subset A ⊆ \mathbbF_q^r. We prove a sieve criterion for existence of such elements, dependent only on an estimate for the character sum ∑_{γ∈ A}χ(γ). The flexibility and direct applicability of our criterion should be of considerable interest for problems in this field. We demonstrate the utility of our result by tackling a problem of Fernandes and Reis (2021) with A avoiding affine hyperplanes, obtaining significant improvements over previous knowledge.

Citations

• Hybrid bounds for prime divisors
• Character sums over elements of extensions of finite fields with restricted coordinates
• Character sums over sparse elements of finite fields
• Primitive elements with prescribed traces
• On primitive elements of finite fields avoiding affine hyperplanes
• Resolving Grosswald's conjecture on GRH
• Approximate formulas for some functions of prime numbers

Cited by

• Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields

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