On primitive and free roots in a finite field
- In this paper the m-dimensional extension IF_(q^m) of the finite field IF_q of order q is investigated from an algebraic point of view. Looking upon the additive group (IF_(q^m), +) as a cyclic module over the principal ideal domain IF_q[x], we introduce a new family of polynomials over IF_q which are the additive analogues of the cyclotomic polynomials. Two methods to calculate these polynomials are proposed. In combination with algorithms to compute cyclotomic polynomials, we obtain, at least theoretically, a method to determine all elements in IF_(q^m) of a given additive and multiplicative order; especially the generators of both cyclic structures, namely the generators of primitive normal bases in IF_(q^m) over IF_q, are characterized as the set of roots of a certain polynomial over IF_q.
| Author: | Dirk HachenbergerORCiDGND |
|---|---|
| URN: | urn:nbn:de:bvb:384-opus4-8232 |
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/967 |
| Parent Title (English): | Applicable Algebra in Engineering, Communications and Computing |
| Type: | Article |
| Language: | English |
| Year of first Publication: | 1992 |
| Publishing Institution: | Universität Augsburg |
| Release Date: | 2008/06/18 |
| Tag: | Finite field; Primitive root; Free root; Normal basis; Primitive normal basis; Cyclotomic polynomial; pi-polynomial |
| GND-Keyword: | Basis <Mathematik>; Galois-Feld; Primitives Element |
| Volume: | 3 |
| Issue: | 2 |
| First Page: | 139 |
| Last Page: | 150 |
| DOI: | https://doi.org/10.1007/BF01387196 |
| 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 |
