Completely normal elements in iterated quadratic extensions of finite fields
type: Item
instance of: scholarly article
title: Completely normal elements in iterated quadratic extensions of finite fields
zbMATH Open document ID: 0952.11028
published in: Finite Fields and their Applications
publication date: 2000-08-14T00:00:00Z
review text: Given a finite field \(F_q\) and an integer \(n \geq 1\), an element \(\alpha\) of \(F_{q^n}\) is called normal over \(F_q\) if its \(F_q\)-conjugates form an \(F_q\)-vector space basis of \(F_{q^n}\). The element \(\alpha\) is called \textit{completely normal over \(F_q\)}, if, simultaneously, for each divisor \(d\) of \(n\), the \(F_{q^d}\)-conjugates of \(\alpha\) constitute an \(F_{q^d}\)-vector space basis of \(F_{q^n}\). For certain ground fields \(F_q\) and certain degrees \(n\) the author provides explicit constructions of (sequences of) irreducible polynomials whose roots are completely normal elements. More precisely, the results are as follows: For \(q \equiv 1\) mod \(4\) and \(n\) a power of \(2\) it is shown that \textit{S. D. Cohen}'s recursive construction of irreducible polynomials of degree \(n\) over \(F_q\) [Des. Codes Cryptography 2, 169-174 (1992; Zbl 0768.11048)] yields polynomials whose roots are completely normal over \(F_q\). The normality of these elements was previously proved by \textit{H. Meyn} [Des. Codes Cryptography 6, 107-116 (1995; Zbl 0834.11053)]. For \(q \equiv 3\) mod \(4\) and \(n\) a power of \(2\) it is shown that McNay's recursive construction of irreducible polynomials of degree \(n\) over \(F_q\) [\textit{G. McNay}, ``Topics in Finite Fields'', Ph.D. thesis, University of Glasgow (1995)] yields polynomials whose roots are completely normal over \(F_q\). The author also discusses a generalization of the polynomial transformations of Cohen and McNay in order to obtain sequences of irreducible polynomials of degree a power of an odd prime \(r\) for ground fields \(F_q\) where \(q \equiv 1\bmod 2r\). Again the roots of these polynomials are completely normal over \(F_q\).
reviewed by: Dirk Hachenberger
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 12E20
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 1199983
zbMATH Keywords: finite field
zbMATH Keywords: Galois field
zbMATH Keywords: normal basis
zbMATH Keywords: complete normal basis
zbMATH Keywords: quadratic extension
author: Robin Chapman
MaRDI profile type: Publication
full work available at URL: ffta.1996.0167
OpenAlex ID: W2026150289
Wikidata QID: Q57918033
cites work: Specific irreducible polynomials with linearly independent roots over finite fields
cites work: Eine Verschärfung des Satzes von der Normalbasis. (An improvement of the normal basis theorem)
cites work: The explicit construction of irreducible polynomials over finite fields
cites work: Explicit N-polynomials of \(2\)-power degree over finite fields. I
DOI: 10.1006/FFTA.1996.0167
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description: scientific article; zbMATH DE number 1199983