A decomposition theory for cyclotomic modules under the complete point of view

  • In 1986, D. Blessenohl and K. Johnsen (1986, J. Algebra 103, 141-159) proved that for any finite extension E/F of Galois fields there exists a complete normal basis generator w of E/F, which means that w simultaneously generates a normal basis for E over every intermediate field of E/F. In a recent monograph by the author (1997, "Finite Fields: Normal Bases and Completely Free Elements," Kluwer Academic, Boston) a theory is developed which allows the study of module structures of Galois fields as extensions with respect to various subfields and which led to an exploration of the structure of complete normal basis generators as well as explicit and algorithmic constructions of these objects. In the present paper we continue the development of that theory by providing various structural results: the Complete Decomposition Theorem, the Complete Product Theorem, a Theorem on Simultaneous Generators, and a Uniqueness Theorem.

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Metadaten
Author:Dirk HachenbergerORCiDGND
URN:urn:nbn:de:bvb:384-opus4-8770
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/1021
Parent Title (English):Journal of Algebra
Type:Article
Language:English
Year of first Publication:2001
Publishing Institution:Universität Augsburg
Release Date:2008/06/20
Tag:Finite field; Galois field; (complete) normal basis; (completely) free/normal element; cyclotomic module; complete/simultaneous generator
GND-Keyword:Galois-Feld; Galois-Erweiterung; Basis <Mathematik>; Kreiskörper
Volume:237
Issue:2
First Page:470
Last Page:486
DOI:https://doi.org/10.1006/jabr.2000.8609
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