On primitive polynomials over finite fields
type: Item
instance of: scholarly article
title: On primitive polynomials over finite fields
zbMATH Open document ID: 0694.12013
DOI: 10.1016/0021-8693(89)90136-1
author: Dieter Jungnickel
author: Scott A. Vanstone
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \(F= \mathrm{GF}(q)\) and \(E=\mathrm{GF}(q^k)\) be the finite fields with \(q\) and \(q^k\) elements, respectively. It is shown that given \(a\in F^*\) there exists a primitive element \(\omega\in E\) satisfying \(T_{E| F}(\omega)=a\) for all but finitely many pairs \((q,k)\). Here \(T_{E| F}(\omega)\) is the trace function of \(E\) with respect to \(F\). Equivalently, the coefficient of \(x^{k-1}\) in a primitive polynomial of degree \(k\) over \(F\) may be arbitrarily selected from \(F^*\). For the case \(k=2\) it is shown the only exceptions to the theorem are for \(q\le 3,847,271\) and there are 147 of them. Some comments on the distribution of the traces of primitive elements are given and in particular the following asymptotic result is shown: if \(q\) sufficiently large then for any \(a\in F^*\) there are at least \(q^{(k-1)(1-\delta)}\) primitive elements of \(E\) with trace a where \(\delta >0\) is arbitrary. For elements of trace zero the following is shown: if \(k\ge 3\) for all but finitely many pairs there exists a primitive element of \(E\) with trace zero over \(F\).
reviewed by: Ian Blake
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 4136074
zbMATH Keywords: distribution of traces
zbMATH Keywords: primitive polynomials
zbMATH Keywords: Gaussian sums
zbMATH Keywords: primitive elements
zbMATH Keywords: asymptotic result
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90136-1
OpenAlex ID: W1996139403
cites work: Q5842121
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description: scientific article; zbMATH DE number 4136074