Primitivity, freeness, norm and trace
- Given the extension E/F of Galois fields, where F = GF(q) and E = GF(q^n), we prove that, for any primitive b element of F*, there exists a primitive element in E which is free over F and whose (E, F)-norm is equal to b. Furthermore, if (q,n) unequal (3,2), we prove that, for any nonzero b element of F, there exists an element in E which is free over F and whose (E,F)-norm is equal to b. A preliminary investigation of the question of determining whether, in searching for a primitive element in E that is free over F, both the (E,F)-norm and the (E,F)-trace can be prescribed is also made: this is so whenever n>=9.
| Author: | Stephen D. Cohen, Dirk HachenbergerORCiDGND |
|---|---|
| URN: | urn:nbn:de:bvb:384-opus4-8754 |
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/1019 |
| Parent Title (English): | Discrete Mathematics |
| Type: | Article |
| Language: | English |
| Year of first Publication: | 2000 |
| Publishing Institution: | Universität Augsburg |
| Release Date: | 2008/06/19 |
| Tag: | Finite field; Primitive element; Free element; Normal basis; Trace; Norm |
| GND-Keyword: | Galois-Feld; Galois-Erweiterung; Spur <Mathematik>; Primitives Element; Basis <Mathematik>; Norm <Mathematik> |
| Volume: | 214 |
| Issue: | 1/3 |
| First Page: | 135 |
| Last Page: | 144 |
| DOI: | https://doi.org/10.1016/S0012-365X(99)00224-1 |
| 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 |
