Quadratic residues and character sums over fields of square order
type: Item
instance of: scholarly article
title: Quadratic residues and character sums over fields of square order
zbMATH Open document ID: 0563.12018
DOI: 10.1016/0022-314X(84)90071-4
author: Stephen D. Cohen
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: For the finite field \(F_ 2=GF(q^ 2)\), where q is an odd prime power, denote by E(k) the set of kth roots of unity in \(F_ 2\). The main objective of this paper is a proof of the following theorem: ''For \(\alpha\) \(\neq 0\) in \(F_ 2\) let \(S(\alpha)=\{\zeta +\alpha:\) \(\zeta \in E((q+1))\}\). Then S(\(\alpha)\) consists entirely of non-squares in \(F_ 2\) if and only if \(q\equiv 3 mod 4\) and \(\alpha \in E(q+1)/E((q+1))\). Further, S(\(\alpha)\) consists entirely of squares in \(F_ 2\) if and only if \(\alpha \in E((q+1))\); these squares exclude 0 if and only if \(q\equiv 1 mod 4''.\) The motivation for this theorem is a construction of M. J. Ganley for a class of weak nucleus semifields leading to a new family of projective planes of order \(q^ 4\). In his proof the author makes use of the modified Jacobsthal sum, \(\sum \chi (x^{2(q-1)}+\alpha)\), where the summation extends over all non-zero x in \(F_ 2\).
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 11L10
Mathematics Subject Classification ID: 12K10
Mathematics Subject Classification ID: 11A15
zbMATH DE Number: 3897068
zbMATH Keywords: finite field
zbMATH Keywords: roots of unity
zbMATH Keywords: non-squares
zbMATH Keywords: weak nucleus semifields
zbMATH Keywords: projective planes of order \(q^ 4\)
zbMATH Keywords: modified Jacobsthal sum
reviewed by: Joseph L. Zemmer
MaRDI profile type: Publication
cites work: Sums of Gauss, Jacobi, and Jacobsthal
cites work: Sums of Gauß, Eisenstein, Jacobi, Jacobsthal, and Brewer
cites work: Pairs of Quadratic Equations in a Finite Field
cites work: Zur Theorie der abstrakten elliptischen Funktionenkörper III. Die Struktur des Meromorphismenrings. Die Riemannsche Vermutung.
description: scientific article; zbMATH DE number 3897068