A generalization of the primitive normal basis theorem
type: Item
instance of: scholarly article
title: A generalization of the primitive normal basis theorem
zbMATH Open document ID: 1207.11119
author: Chih-Nung Hsu
author: Ting-Ting Nan
published in: Journal of Number Theory
publication date: 2010-11-19T00:00:00Z
review text: Let \(A=\mathbb F_q[T]\) and fix irreducible \(f\in A\) of degree \(d\). Set \(E_m=\mathbb F_{q^{dm}}\). One can define an \(A\)-module structure on \(E_m\) resulting in the Carlitz module \(C(E_m)\). \(C(E_m)\) is isomorphic, as an \(A\)-module, to \(A/(f^m-1)\) and so is cyclic. A primitive generator for \(C(E_m)\) is a primitive element \(\alpha\in E_m\) that generates \(C(E_m)\). The authors prove that \(C(E_m)\) has a primitive generator except possibly for finitely many \((q, d, m)\), which are given explicitly. They note there is no primitive generator for \((q,d,m)=(2,2,1)\). When \(f=T\), a primitive generator of \(C(E_m)\) yields a primitive normal basis for \(E_m\) over \(E_1\). Lenstra and Schoof have shown that primitive normal bases exist in all cases.
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 5817571
zbMATH Keywords: character sum
zbMATH Keywords: primitive root
zbMATH Keywords: Carlitz module
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.05.011
OpenAlex ID: W2000389682
cites work: Primitive Roots in a Finite Field
cites work: Q5814878
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cites work: Bases for Finite Fields
cites work: Q4714153
cites work: Primitive Normal Bases for Finite Fields
DOI: 10.1016/J.JNT.2010.05.011
description: scientific article; zbMATH DE number 5817571