Generators for primary closures of Galois fields.
type: Item
instance of: scholarly article
title: Generators for primary closures of Galois fields.
zbMATH Open document ID: 1075.11080
DOI: 10.1016/S1071-5797(02)00021-7
author: Dirk Hachenberger
published in: Finite Fields and their Applications
publication date: 2003-04-02T00:00:00Z
review text: Let \(q\) be a prime power, \(k\) and \(e\) positive integers and \((\mathbb F_q,\mathbb F_{q^k},\mathbb F_{q^{ek}})\) a tower of finite fields of order \(q\), \(q^k\) and \(q^{ek}\), respectively. Let \({\mathcal F}\) denote the set of triples \((q,k,e)\) for which, for every \(a\in \mathbb F_{q^k}\) normal over \(\mathbb F_q\) (i.e.\ \(\{a,a^q,\dots,a^{q^{k-1}}\}\) is a basis of \(\mathbb F_{q^k}\)), there exists a primitive element \(\gamma \in \mathbb F_{q^{ek}}\), normal over \(\mathbb F_q\), with \(\mathbb F_{q^k}\)-trace equal to \(a\). Theorem 1.1.: Let \(r\) be an odd prime different from the characteristic of \(\mathbb F_q\), \(s\) the order of \(q\) modulo \(r\) and assume that the order of \(q\) modulo \(r^2\) is equal to \(rs\). Then \((q,r^a,r^b)\in {\mathcal F}\) for all \(a\geq 0\) and \(b\geq 1\). This complements earlier results of the author [Finite Fields Appl. 5, No. 4, 378--385 (1999; Zbl 0982.11070)]. A quadruple \((q,k,l,n)\) with a prime power \(q\) and positive integers \(k,l,n\) such that \(k\) and \(l\) both divide \(n\) is called \textit{universal} if for any \(a\in \mathbb F_{q^k}\), normal over \(\mathbb F_q\), and any primitive element \(b\) of \(\mathbb F_{q^l}\), there exists a primitive element \(\gamma\) of \(\mathbb F_{q^n}\), normal over \(\mathbb F_q\), with \(\mathbb F_{q^k}\)-trace equal to \(a\) and \(\mathbb F_{q^l}\)-trace equal to \(b\). Theorem 1.2: Let \(r\geq 5\) be a prime different from the characteristic of \(\mathbb F_q\), \(s\) the order of \(q\) modulo \(r\) and assume that the order of \(q\) modulo \(r^2\) is equal to \(rs\). Then \((q,r^a,r^b,r^c)\) is a universal quadruple for all \(a,b\geq 0\) and all \(c>\max(a,b)\). This complements the author's earlier results in [Universal generators for primary closures of Galois fields, in Finite fields and applications. Proceedings of the fifth international conference on finite fields and applications \(F{_q^5}\), University of Augsburg, Germany, August 2--6, 1999, Be…
reviewed by: Arne Winterhof
Mathematics Subject Classification ID: 11T30
zbMATH DE Number: 1891478
zbMATH Keywords: finite (Galois) field
zbMATH Keywords: primitive element
zbMATH Keywords: normal (free) element
zbMATH Keywords: normal basis
zbMATH Keywords: completely normal (completely free) element
zbMATH Keywords: trace
zbMATH Keywords: norm
zbMATH Keywords: trace-compatible sequence
zbMATH Keywords: norm-compatible sequence
zbMATH Keywords: universal quadruple
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1891478