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An element α of the extension E of degree n over the finite field F = GF (q) is called free over F if {α, α q , . . . , α 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 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 .

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