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On the minimal polynomials for certain Gauss periods over finite fields. (English) Zbl 0874.11080

Cohen, S. (ed.) et al., Finite fields and applications. Proceedings of the 3rd international conference, Glasgow, UK, July 11–14, 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 233, 85-96 (1996).
Summary: Denote by \(\mathbb{F}_q\) the finite field of \(q\) elements, where \(q=ef+1\) is a power of a prime \(p\). Set \(\delta=\text{gcd}\left({q-1\over p-1},e\right)\). G. Myerson showed that the period polynomial \(\Phi(x)\) of degree \(e\) for the finite field \(\mathbb{F}_q\) factors over the rational field \(\mathbb{Q}\) as a product of \(\delta\) polynomials, \(\Phi^{(\omega)}(x)\) \((1\leq\omega\leq\delta)\), each of degree \(e/\delta\). Focusing on the last factor \(\Phi^{(\delta)}(x)\) with \(f\) fixed, the author recently obtained formulas for the beginning coefficients of \(\Phi^{(\delta)}(x)\) expressed in terms of a certain counting function \(\beta_K(n)\) for positive integers \(n\), where \(K\) is the decomposition field of \(p\) in the cyclotomic field of \(f\)-roots of unity. These formulas are computationally impractical since \(\beta_K(n)\) is difficult to compute, in general. However, he has found closed form expressions for generating functions for \(\beta_K(n)\) when \(K=\mathbb{Q}\), which lead to explicit expressions for the beginning coefficients of \(\Phi^{(\delta)}(x)\).
For the entire collection see [Zbl 0851.00052].

MSC:

11T22 Cyclotomy
11T24 Other character sums and Gauss sums
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