Given
$q$
, a power of a prime
$p$
, denote by
$F$
the finite field
${\rm GF}(q)$
of order
$q$
, and, for a given positive integer
$n$
, by
$E$
its extension
${\rm GF}(q^n)$
of degree
$n$
. A primitive element of
$E$
is a generator of the cyclic group
$E^\ast$
. Additively too, the extension
$E$
is cyclic when viewed as an
$FG$
-module,
$G$
being the Galois group of
$E$
over
$F$
. The classical form of this result – the normal basis theorem – is that there exists an element
$\alpha \in E$
(an additive generator) whose conjugates
$\{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$
form a basis of
$E$
over
$F; \alpha$
is a free element of
$E$
over
$F$
, and a basis like this is a normal basis over
$F$
. The core result linking additive and multiplicative structure is that there exists
$\alpha \in E$
, simultaneously primitive and free over
$F$
. This yields a primitive normal basis over
$F$
, all of whose members are primitive and free. Existence of such a basis for every extension was demonstrated by Lenstra and Schoof [5] (completing work by Carlitz [1, 2] and Davenport [4]).