Primitive normal bases with prescribed trace
- Let E be a finite degree extension over a finite field F = GF(q), G the Galois group of E over F and let a element of F be nonzero. We prove the existence of an element w in E satisfying the following conditions: (1) W is primitive in E, i.e., W generates the multiplicative group of E (as a module over the ring of integers). (2) the set {w^g I g element of G} of conjugates of w under G forms a normal basis of E over F. (3) the (E, F)-trace of w is equal to a. This result is a strengthening of the primitive normal basis theorem of Lenstra and Schoof [10] and the theorem of Cohen on primitive elements with prescribed trace [3]. It establishes a recent conjecture of Morgan and Mullen [14], who, by means of a computer search, have verified the existence of such elements for the cases in which q <= 97 and n <= 6, n being the degree of E over F. Apart from two pairs (F, E) (or (q, n)) we are able to settle the conjecture purely theoretically.
Author: | Stephen D. Cohen, Dirk HachenbergerORCiDGND |
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URN: | urn:nbn:de:bvb:384-opus4-8612 |
Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/1005 |
Parent Title (English): | Applicable Algebra in Engineering, Communication and Computing |
Type: | Article |
Language: | English |
Year of first Publication: | 1999 |
Publishing Institution: | Universität Augsburg |
Release Date: | 2008/06/19 |
Tag: | Finite field; Primitive element; Normal basis; Free element; Trace; Character sum |
GND-Keyword: | Galois-Feld; Spur <Mathematik>; Primitives Element; Basis <Mathematik> |
Volume: | 9 |
Issue: | 5 |
First Page: | 383 |
Last Page: | 403 |
DOI: | https://doi.org/10.1007/s002000050112 |
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 |