On completely free elements in finite fields

  • Let q > 1 be a prime power, m > 1 an integer, GF(q^m) and GF(q) the Galois fields of order q^m and q, respectively. We show that the different module structures of (GF(q^m), +) arising from the intermediate fields of the field extension GF(q^m) over GF(q) can be studied simultaneously with the help of some basic properties of cyclotomic polynomials. The results can be generalized to finite cyclic Galois extensions over arbitrary fields. In 1986, D. Blessenohl and K. Johnsen proved that there exist elements in GF(q^m) which generate normal bases in GF(q^m) over any intermediate field GF(q^d) of GF(q^m) over GF(q). Such elements are called completely free in GF(q^m) over GF(q). Using our ideas, we give a detailed and constructive proof of the most difficult part of that theorem, i.e., the existence of completely free elements in GF(q^m) over GF(q) provided that m is a prime power. The general existence problem of completely free elements is easily reduced to this special case.Let q > 1 be a prime power, m > 1 an integer, GF(q^m) and GF(q) the Galois fields of order q^m and q, respectively. We show that the different module structures of (GF(q^m), +) arising from the intermediate fields of the field extension GF(q^m) over GF(q) can be studied simultaneously with the help of some basic properties of cyclotomic polynomials. The results can be generalized to finite cyclic Galois extensions over arbitrary fields. In 1986, D. Blessenohl and K. Johnsen proved that there exist elements in GF(q^m) which generate normal bases in GF(q^m) over any intermediate field GF(q^d) of GF(q^m) over GF(q). Such elements are called completely free in GF(q^m) over GF(q). Using our ideas, we give a detailed and constructive proof of the most difficult part of that theorem, i.e., the existence of completely free elements in GF(q^m) over GF(q) provided that m is a prime power. The general existence problem of completely free elements is easily reduced to this special case. Furthermore, we develop a recursive formula for the number of completely free elements in GF(q^m) over GF(q) in the case where m is a prime power.show moreshow less

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Metadaten
Author:Dirk HachenbergerORCiDGND
URN:urn:nbn:de:bvb:384-opus4-8400
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/984
Parent Title (English):Designs, Codes and Cryptography
Type:Article
Language:English
Year of first Publication:1994
Publishing Institution:Universität Augsburg
Release Date:2008/06/18
Tag:free element; finite fields; Galois field; field extension; normal bases
GND-Keyword:Galois-Feld; Galois-Erweiterung; Basis <Mathematik>
Volume:4
Issue:2
First Page:129
Last Page:143
DOI:https://doi.org/10.1007/BF01578867
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Diskrete Mathematik, Optimierung und Operations Research
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik