Chang, Mei-Chu Order of Gauss periods in large characteristic. (English) Zbl 1344.11078 Taiwanese J. Math. 17, No. 2, 621-628 (2013). Summary: Let \(p\) be the characteristic of \(\mathbb F_q\) and let \(q\) be a primitive root modulo a prime \(r = 2n + 1\). Let \(\beta\in\mathbb F_{q^{2n}}\) be a primitive \(r\)th root of unity. We prove that the multiplicative order of the Gauss period \(\beta+\beta^{-1}\) is at least \((\log p)^{c\log n}\) for some \(c>0\). This improves the bound obtained by Ahmadi, Shparlinski and Voloch [O. Ahmadi et al., Int. J. Number Theory 6, No. 4, 877–882 (2010; Zbl 1201.11110)] when \(p\) is very large compared with \(n\). We also obtain bounds for ”most” \(p\). Cited in 5 Documents MSC: 11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects) 11B30 Arithmetic combinatorics; higher degree uniformity 11G20 Curves over finite and local fields 12E20 Finite fields (field-theoretic aspects) 14G15 Finite ground fields in algebraic geometry Keywords:multiplicative order; multiplicative group; finite fields; additive combinatorics Citations:Zbl 1201.11110 PDFBibTeX XMLCite \textit{M.-C. Chang}, Taiwanese J. Math. 17, No. 2, 621--628 (2013; Zbl 1344.11078) Full Text: DOI Link