Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T14:45:29.003Z Has data issue: false hasContentIssue false

THE PRIMITIVE NORMAL BASIS THEOREM – WITHOUT A COMPUTER

Published online by Cambridge University Press:  25 March 2003

STEPHEN D. COHEN
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW [email protected]
SOPHIE HUCZYNSKA
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW [email protected]
Get access

Abstract

Given $q$ , a power of a prime $p$ , denote by $F$ the finite field ${\rm GF}(q)$ of order $q$ , and, for a given positive integer $n$ , by $E$ its extension ${\rm GF}(q^n)$ of degree $n$ . A primitive element of $E$ is a generator of the cyclic group $E^\ast$ . Additively too, the extension $E$ is cyclic when viewed as an $FG$ -module, $G$ being the Galois group of $E$ over $F$ . The classical form of this result – the normal basis theorem – is that there exists an element $\alpha \in E$ (an additive generator) whose conjugates $\{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ form a basis of $E$ over $F; \alpha$ is a free element of $E$ over $F$ , and a basis like this is a normal basis over $F$ . The core result linking additive and multiplicative structure is that there exists $\alpha \in E$ , simultaneously primitive and free over $F$ . This yields a primitive normal basis over $F$ , all of whose members are primitive and free. Existence of such a basis for every extension was demonstrated by Lenstra and Schoof [5] (completing work by Carlitz [1, 2] and Davenport [4]).

Type
Notes and Papers
Copyright
The London Mathematical Society, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)