  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.
