Primitive polynomials over finite fields of characteristic two
type: Item
instance of: scholarly article
title: Primitive polynomials over finite fields of characteristic two
zbMATH Open document ID: 1056.11077
author: Shuqin Fan
author: Wenbao Han
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2004-09-22T00:00:00Z
review text: According to a conjecture of \textit{T. Hansen} and \textit{G. L. Mullen} [Math. Comput. 59, 639--643 (1992; Zbl 0770.11053)], for any given element \(a\in\mathbb F_q\) and integers \(n\) and \(m\) there exists a primitive polynomial of degree \(n\) over \(\mathbb F_q\) whose \(m\)th coefficient is equal to \(a\), except for three sporadic cases. In [Proc. Am. Math. Soc. 132, No. 1, 15--31 (2004; Zbl 1047.11119)], the authors showed that the conjecture is true for sufficiently large \(q\) (depending on \(n\)) and \(m\not=\frac{n+1}{2}\) or \(m\not=\frac{n}{2},\frac{n}{2}+1\). Now they prove the existence of primitive polynomials with an arbitrary prescribed coefficient over fields of characteristic two for odd values of \(n\geq7\) whatever the size of \(q\). The existence problem of primitive polynomials over finite fields is translated into the existence problem of primitive element solutions of systems of trace equations over Galois rings. Then character sum methods and Cohen's sieve yield conditions for the solvability of such systems.
reviewed by: Astrid Reifegerste
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 11T24
zbMATH DE Number: 2101931
zbMATH Keywords: primitive polynomials over finite fields
zbMATH Keywords: Hansen-Mullen conjecture
zbMATH Keywords: Galois rings
zbMATH Keywords: character sums over Galois rings
zbMATH Keywords: Cohen's sieve method
MaRDI profile type: Publication
full work available at URL: s00200-003-0140-6
OpenAlex ID: W2029298180
DOI: 10.1007/S00200-003-0140-6
description: scientific article; zbMATH DE number 2101931