Specific irreducible polynomials with linearly independent roots over finite fields
type: Item
instance of: scholarly article
title: Specific irreducible polynomials with linearly independent roots over finite fields
zbMATH Open document ID: 0870.11075
DOI: 10.1016/0024-3795(95)00744-X
author: Ian Blake
author: Shuhong Gao
author: Ronald C. Mullin
published in: Linear Algebra and its Applications
publication date: 1997-05-13T00:00:00Z
review text: If \(F=\text{GF}(q)\) is a finite field and \(n\geq 1\) an integer, then it is one of the major problems in the theory of finite fields to (explicitly) construct an irreducible polynomial of degree \(n\) over \(F\) (possibly satisfying additional properties such as normality, primitivity or prescribed coefficients). The authors determine three families of so-called completely normal polynomials. (A monic polynomial \(f \in\) GF\((q)\) of degree \(n\) is completely normal if \(f\) is irreducible and if every root \(\alpha\) of \(f\) simultaneously generates a normal basis for GF\((q^n)\) over every intermediate field of GF\((q^n)\) over GF\((q)\).) Completely normal (or completely free) elements have attained much interest. In this respect, we would like to mention the following references (the second one is a monograph on simultaneous module structures over finite fields). [1] \textit{R. Chapman} [Finite Fields Appl. 3, 1-10 (1997);]. [2] \textit{D. Hachenberger}, Finite fields: Normal bases and completely free elements, Kluwer Academic Publishers, Boston (1997; Zbl 0864.11065). The degrees of the polynomials determined in the paper under review depend strongly on properties of the given ground field. The first construction works for all \((n,q)\) in which there exists an irreducible binomial of degree \(n\) over GF\((q)\) (in fact binomials play a crucial role for the construction). The second construction works for all pairs \((2^{k+1},q)\), \(k \geq 0\), \(q-3\) divisible by \(4\), while the third family has parameters \((p^{k+1},p)\), \(k \geq 0\), \(p\) prime. Compared to the first and the third family, the proof concerning the second one is very involved and demands more than half of the paper. In fact it is the normality over the ground field rather than the completeness which causes the trouble. The (iterative) construction in [1] of a family of completely normal polynomials with parameters \((2^{k+1},q)\), \(k \geq 0\), \(q-3\) divisible by \(4\) allows a s…
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 994642
zbMATH Keywords: irreducible polynomials with linearly independent roots over finite fields
zbMATH Keywords: completely normal polynomials
zbMATH Keywords: completely free elements
zbMATH Keywords: binomials
reviewed by: Dirk Hachenberger
MaRDI profile type: Publication
cites work: Low complexity normal bases
cites work: Explicit factorization of \(x^{2^ k}+1\) over \(F_ p\) with prime \(p\equiv 3\bmod 4\)
cites work: Normal and Self-Dual Normal Bases from Factorization of $cx^{q + 1} + dx^q - ax - b$
cites work: Supplement zu ``Eine Verschärfung des Satzes von der Normalbasis''. (Supplement to `A stronger form of the normal basis theorem')
cites work: Eine Verschärfung des Satzes von der Normalbasis. (An improvement of the normal basis theorem)
cites work: Optimal normal bases
cites work: Constructing normal bases in finite fields
cites work: On completely free elements in finite fields
cites work: Q5614003
cites work: Finding Isomorphisms Between Finite Fields
cites work: Q3216750
cites work: Q3809926
cites work: Q3136478
cites work: Optimal normal bases in \(GF(p^ n)\)
cites work: Normal basis of finite field&lt;tex&gt;GF(2^m)&lt;/tex&gt;(Corresp.)
cites work: Normal bases of cyclic fields of primepower degree
cites work: Bases for finite fields and a canonical decomposition for a normal basis generator
cites work: Q4314377
cites work: Q3798760
cites work: CONSTRUCTION OF POLYNOMIALS IRREDUCIBLE OVER A FINITE FIELD WITH LINEARLY INDEPENDENT ROOTS
cites work: ON NORMAL BASES OF A FINITE FIELD
cites work: О построении нормального базиса конечного поля
cites work: Q3724438
cites work: Explicit iterative constructions of normal bases and completely free elements in finite fields
cites work: Normal bases and completely free elements in prime power extensions over finite fields
description: scientific article; zbMATH DE number 994642