Explicit lifts of quintic Jacobi sums and period polynomials for \(\mathbb F_q\)
type: Item
instance of: scholarly article
title: Explicit lifts of quintic Jacobi sums and period polynomials for \(\mathbb F_q\)
zbMATH Open document ID: 1116.11093
author: Akinari Hoshi
published in: Proceedings of the Japan Academy. Series A
publication date: 2007-06-06T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1160485503
review text: From the author's introduction: Let \(e\geq 2\) be a positive integer and \(q=p^r\) a prime power such that \(q\equiv 1\pmod e\). Write \(q=ef+1\). Let \(\zeta_p\) be a \(p\)th primitive root of unity, \(\gamma\) a fixed generator of \(\mathbb F_q\). Gaussian periods \(\eta_{0,r},\dots, \eta_{e-1,r}\) of degree \(e\) for \(\mathbb F_q\) are defined by \[ \eta_{i,r}:= \sum_{j=0}^{f-1} \zeta_p^{\text{Tr} (\gamma^{ej+i})}, \] where GR is the trace map \(\text{Tr}:\mathbb F_q\to\mathbb F_p\), and the period polynomial \(P_{e.r}(X)\) of degree \(e\) for \(\mathbb F_q\) is given by \(P_{e,r}(X):= \prod_{i=0}^{e-1} (X-\eta_{i,r})\). We also use the reduced form \(P_{e,r}^*(X):= \prod_{i=0}^{e-1} (X-\eta_{i,r}^*)\), where \(\eta_{i,r}^*= e\eta_{i,r}+1\), since the coefficient of \(X^{e-1}\) of \(P_{e,r}^*(X)\) is vanished. In the classical case \(q=p\), \textit{C. F. Gauss} [Disquisitiones Arithmeticae, Section 358] showed that the period polynomial \(P_{e,1}(X)\) is irreducible over \(\mathbb Q\). However this is not always true for general \(q=p^r\). In 1981, for \(\delta= \gcd(e,(q-1)/(p-1))\), \textit{G. Myerson} [ Acta Arith. 39, 251--264 (1981; Zbl 0393.12028)] showed that the period polynomial \(P_{e,r}(X)\) splits over \(\mathbb Q\) into \(\delta\) factors \[ P_{e,r}(X)= \prod_{k=0}^{\delta-1} P_{e,r}^{(k)}(X), \] where \(P_{e,r}^{(k)}(X)\) is in \(\mathbb Z[X]\) and irreducible or a power of an irreducible polynomial. Note that \(P_{e,r}(X)\) is irreducible over \(\mathbb Q\) if and only if \(p\equiv 1\pmod e\) and \((r,e)= 1\), i.e. \(\delta=1\). The explicit determination of the factors of \(P_{e,r}(X)\), if reducible, is important because it is known that the (exponential) Gauss sum \(g_r(e)\) is one of the roots of \(P_{e,r}^*(X)\) [see \textit{B. C. Berndt, R. J. Evans} and \textit{K. S. Williams}, Gauss and Jacobi sums, New York, NY: John Wiley \& Sons (1998; Zbl 0906.11001)]. Myerson (loc. cit.) determined the factors \(P_{e,r}^{(k)}(X)\) for \(e=2,3,4\). In…
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11T22
zbMATH DE Number: 5161646
zbMATH Keywords: Jacobi sums
zbMATH Keywords: Gaussian periods
zbMATH Keywords: Dickson's system
zbMATH Keywords: Gauss sums
reviewed by: Olaf Ninnemann
describes a project that uses: Mathematica
MaRDI profile type: Publication
OpenAlex ID: W2076272501
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DOI: 10.3792/PJAA.82.87
description: scientific article; zbMATH DE number 5161646