A note on integral matrices associated to finite fields
type: Item
instance of: scholarly article
title: A note on integral matrices associated to finite fields
zbMATH Open document ID: 0644.12009
DOI: 10.1016/0021-8693(88)90305-5
author: Jacob Nemchenok
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathbb{F}}_ q\) be the finite field of q elements, \(q\equiv 1 (e)\), e a fixed integer, \(H_ f\) the unique subgroup of \({\mathbb{F}}^*_ q\) of order \(f=(q-1)/e\), g a generator of \({\mathbb{F}}^*_ q\), \(c_{ij}=card(H_ f\cap (g^ i+g^ jH_ f))\) (0\(\leq i,j\leq e-1)\) and \(c=c(q)=\min_{i,j}(c_{ij}).\) \textit{E. Snapper} [J. Algebra 97, 267-277 (1985; Zbl 0573.12007)] conjectured that \(\lim_{q\to \infty}c(q)=\infty\), where q ranges over primes \(\equiv 1 (mod e)\). - By using a well known lower bound for the number of solutions of \(ax^ e+by^ e=c\) over \({\mathbb{F}}_ q\), the author proves Snapper's conjecture where q ranges over prime powers\(\equiv 1 (mod e)\).
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11C20
zbMATH DE Number: 4049748
zbMATH Keywords: subgroup of finite field
reviewed by: A. R. Rajwade
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90305-5
OpenAlex ID: W2017651523
cites work: Finite fields, integral matrices, and diophantine equations
cites work: Number of Points of Varieties in Finite Fields
description: scientific article; zbMATH DE number 4049748