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Order of Gauss periods in large characteristic. (English) Zbl 1344.11078

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” \(p\).

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

Citations:

Zbl 1201.11110
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