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]).