The Strong Primitive Normal Basis Theorem

2006/10/12 by Cohen, Stephen D., Huczynska, Sophie · 1 citation
#11T30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0610400

Abstract

An element w of the extension E of degree n over the finite field F=GF(q) is called free over F if w, w^q,...,w^q^n-1 is a (normal) basis of E/F. The Primitive Normal Basis Theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F, there exists an element w in E such that w is simultaneously primitive (i.e., generates the multiplicative group of E) and free over F. In this paper we prove the following strengthening of this theorem: aside from five specific extensions E/F, there exists an element w in E such that both w and w^-1 are simultaneously primitive and free over F.

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