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.
Author: | Dirk Hachenberger |
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URN: | urn:nbn:de:hbz:386-kluedo-50424 |
Series (Serial Number): | Preprints (rote Reihe) des Fachbereich Mathematik (232) |
Document Type: | Report |
Language of publication: | English |
Date of Publication (online): | 2017/11/07 |
Year of first Publication: | 1992 |
Publishing Institution: | Technische Universität Kaiserslautern |
Date of the Publication (Server): | 2017/11/07 |
Page Number: | 15 |
Faculties / Organisational entities: | Kaiserslautern - 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) |