Normal basis theorem

Let $E$ be a (finite-dimensional) Galois extension of a field $F$. Then there exists a normal basis for $E/F$, that is, a basis consisting of an orbit of the Galois group $G = \mathrm{Gal}(E/F)$. Thus, an element $z \in E$ generates a normal basis if and only if its conjugates $z^\sigma$, $\sigma \in G$, are linearly independent over $F$; see, e.g., [a3]. The element $z$ is called a normal basis generator or a free element in $E/F$. A far-reaching strengthening of the normal basis theorem is due to D. Blessenohl and K. Johnsen [a1]: There exists an element $w \in E$ that is simultaneously free in $E/K$ for every intermediate field $K$.
For the important special case where $E$ is a Galois field, a constructive treatment of normal bases and completely free elements can be found in [a2]. In this case, there is always a normal basis consisting of primitive elements (elements of maximal multiplicative order, cf. Primitive element of a Galois field), see [a4]