Primitive normal bases for towers of field extensions
type: Item
instance of: scholarly article
title: Primitive normal bases for towers of field extensions
zbMATH Open document ID: 0982.11070
author: Dirk Hachenberger
published in: Finite Fields and their Applications
publication date: 2001-09-12T00:00:00Z
full work available at URL: https://opus.bibliothek.uni-augsburg.de/opus4/frontdoor/index/index/docId/1006
review text: Let \(q\) be a prime power, \(k,e\) positive integers, and \(F,K,E\) the finite fields of order \(q,q^k,q^{ek}\), respectively. Let \({\mathcal F}\) be the set of triples \((q,k,e)\) such that, for every \(a\in K\) which is normal over \(F\), there exists a primitive \(w_a\) in \(E\) which is normal over \(F\), and whose \((E,K)\)-trace equals \(a\). \textit{S. D. Cohen} and the author [Appl. Algebra Eng. Commun. Comput. 9, 383-403 (1999; Zbl 0931.12002)] proved that \((q,1,e)\in {\mathcal F}\) for all \(e\geq 2\) and all \(q\). In the paper under review the author considers the case \(k>1\) and proves the following results: 1. \((q,k,e)\in {\mathcal F}\) for all \(e\geq 2\) divisible by the characteristic of \(F\), all \(k\), and all \(q\). 2. There are at most finitely many triples \((q,k,e)\) with \(q\geq 17\), \(k\geq 1\), and \(e\geq 3\) with \((q,k,e)\not\in {\mathcal F}\). 3. \((q,k,e)\in {\mathcal F}\) for \(k\) and \(e\) powers of the same prime \(r\geq 5\) and \((q,2^{\alpha+3},2^\beta),(q,3^\alpha,3^\beta)\in {\mathcal F}\) for all \(\alpha\geq 0\) and \(\beta\geq 2\) for all \(q\). The last result implies the existence of trace-compatible sequences of primitive normal elements in towers of finite fields with constant extension degree \(r\), where \(r\geq 5\) is prime.
reviewed by: Arne Winterhof
Mathematics Subject Classification ID: 11T30
Mathematics Subject Classification ID: 12E20
zbMATH DE Number: 1369828
zbMATH Keywords: finite fields
zbMATH Keywords: trace
zbMATH Keywords: normal basis
zbMATH Keywords: primitive element
zbMATH Keywords: towers of finite fields
MaRDI profile type: Publication
OpenAlex ID: W1978498901
cites work: Primitive elements and polynomials with arbitrary trace
cites work: Primitive normal bases with prescribed trace
cites work: Finite fields: normal bases and completely free elements
cites work: Q2712112
cites work: Q3243354
cites work: Primitive Normal Bases for Finite Fields
cites work: Primitive Normal Polynomials Over Finite Fields
DOI: 10.1006/FFTA.1999.0256
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description: scientific article; zbMATH DE number 1369828