The Strong Primitive Normal Basis Theorem
Stephen D. Cohen, Sophie Huczynska
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, wq,...,wqn-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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