The strong primitive normal basis theorem
Stephen D. Cohen, Sophie Huczynska · Acta Arithmetica · 2010
An element α of the extension E of degree n over the finite field F = GF(q) is called free over F if {α, αq,...,α qn−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 α ∈ E such that α 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 α ∈ E such that both α and α −1 are simultaneously primitive and free over F. 1 1