On elements of high order in general finite fields

• Date: 2018
• Main Author: Popovych, Roman
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2018
• Subjects: finite field; multiplicative order; Diophantine inequality; 11T30
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1062
• Journal Title: Algebra and Discrete Mathematics

We show that the Gao's construction gives for any finite field \(F_{q^{n}}\) elements with the multiplicative order at least \(\binom{n+t-1}{t}\prod _{i=0}^{t-1}\frac{1}{d^{i}}\), where \(d=\left\lceil 2\log _{q} n\right\rceil\), \(\;t=\left\lfloor \log _{d} n\right\rfloor\).