Explicit iterative constructions of normal bases and completely free elements in finite fields
- A characterization of normal bases and complete normal bases in GF(q^(r^n)) over GF(q), where q > 1 is any prime power, r is any prime number different from the characteristic of GF(q), and n >= 1 is any integer, leads to a general construction scheme of series (v_n)n>=0 in GF(q^(r^infinite) having the property that the partial Sums w_n are free or completely free in GF(q^(r^n)) over GF(q), depending on the choice of v_n. In the case where r is an odd prime divisor of q - 1 or where r = 2 and q = 1 mod 4, for any integer n >= 1, all free and completely free elements in GF(q^(r^n)) over GF(q) are explicitly determined in terms of certain roots of unity. In the case where r = 2 and q = 3 mod 4, for any n >= 1, in terms of certain roots of unity, an explicit recursive construction for free and completely free elements in GF(q^(2^n)) over GF(q) is given. As an example, for a particular series of completely free elements the corresponding minimal polynomials are given explicitly.
Author: | Dirk HachenbergerORCiDGND |
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URN: | urn:nbn:de:bvb:384-opus4-8422 |
Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/986 |
Parent Title (English): | Finite Fields and Their Applications |
Type: | Article |
Language: | English |
Year of first Publication: | 1996 |
Publishing Institution: | Universität Augsburg |
Release Date: | 2008/06/19 |
Tag: | normal bases; free elements; completely free elements; finite fields |
GND-Keyword: | Galois-Feld; Basis <Mathematik> |
Volume: | 2 |
Issue: | 1 |
First Page: | 1 |
Last Page: | 20 |
DOI: | https://doi.org/10.1006/ffta.1996.0001 |
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 |