Primitive complete normal bases for regular extensions
- The extension E of degree n over the Galois field F = GF(q) is called regular over F, if ord_r(q) and n have greatest common divisor 1 for all prime divisors r of n which are different from the characteristic p of F (here, ord_r(q) denotes the multiplicative order of q modulo r). Under the assumption that E is regular over F and that q - 1 is divisible by 4 if q is odd and n is even, we prove the existence of a primitive element w element of E which is also completely normal over F (the latter means that w simultaneously generates a normal basis for E over every intermediate field K of E/F). Our result achieves, for the class of extensions under consideration, a common generalization of the theorem of Lenstra and Schoof on the existence of primitive normal bases [12] and the theorem of Blessenohl and Johnsen on the existence of complete normal bases [1].
| Author: | Dirk HachenbergerORCiDGND |
|---|---|
| URN: | urn:nbn:de:bvb:384-opus4-8789 |
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/1022 |
| Parent Title (English): | Glasgow Mathematical Journal |
| Type: | Article |
| Language: | English |
| Year of first Publication: | 2001 |
| Publishing Institution: | Universität Augsburg |
| Release Date: | 2008/06/20 |
| Tag: | normal bases; Galois field; regular extensions; primitive element; free element; generator |
| GND-Keyword: | Galois-Feld; Galois-Erweiterung; Basis <Mathematik>; Primitives Element |
| Volume: | 43 |
| Issue: | 3 |
| First Page: | 383 |
| Last Page: | 398 |
| DOI: | https://doi.org/10.1017/S0017089501030026 |
| 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 |
