On Completely Free Elements in Finite Fields

  • We show that the different module structures of GF(\(q^m\)) arising from the intermediate fields of GF(\(q^m\))and GF(q) can be studied simultaneously with the help of some basic properties of cyclotomic polynomials. We use this ideas to give a detailed and constructive proof of the most difficult part of a Theorem of D. Blessenohl and K. Johnsen (1986), i.e., the existence of elements v in GF(\(q^m\)) over GF(q) which generate normal bases over any intermediate field of GF(\(q^m\)) and GF(q), provided that m is a prime power. Such elements are called completely free in GF(\(q^m\)) over GF(q). 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. Some of the results can be generalized to finite cyclic Galois extensions over arbitrary fields.

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Author:Dirk Hachenberger
URN (permanent link):urn:nbn:de:hbz:386-kluedo-50424
Serie (Series number):Preprints (rote Reihe) des Fachbereich Mathematik (232)
Document Type:Report
Language of publication:English
Publication Date:2017/11/07
Year of Publication:1992
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2017/11/07
Number of page:15
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0)