Normal bases and completely free elements in prime power extensions over finite fields
type: Item
instance of: scholarly article
title: Normal bases and completely free elements in prime power extensions over finite fields
zbMATH Open document ID: 0858.11063
author: Dirk Hachenberger
published in: Finite Fields and their Applications
publication date: 1996-05-08T00:00:00Z
full work available at URL: https://opus.bibliothek.uni-augsburg.de/opus4/frontdoor/index/index/docId/993
review text: For \(q\) any prime power, \(m\) any positive integer, an element \(w\) of the Galois field \(GF(q^m)\) is free (resp., completely free) over \(GF(q)\) provided it generates a normal basis in \(GF(q^m)\) over \(GF(q)\) (resp., it simultaneously generates normal bases in \(GF(q^m)\) over each intermediate field \(GF(q^d)\) of \(GF(q^m)\) over \(GF(q))\). Continuing his earlier work [Finite Fields Appl. 2, No. 1, 1-20 (1996; Zbl 0848.11061)], the author is able to explicitly determine free and completely free elements in \(GF(q^m)\) over \(GF(q)\) for every nonnegative integer \(m\) and every prime power \(q\). Moreover, in many cases he is able to describe all completely free elements in terms of a certain primitive root of unity.
reviewed by: Stanley E. Payne
Mathematics Subject Classification ID: 11T30
Mathematics Subject Classification ID: 12E20
zbMATH DE Number: 851859
zbMATH Keywords: prime power extensions over finite fields
zbMATH Keywords: free elements
zbMATH Keywords: normal basis
zbMATH Keywords: completely free elements
MaRDI profile type: Publication
OpenAlex ID: W2048665286
DOI: 10.1006/FFTA.1996.0002
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description: scientific article; zbMATH DE number 851859