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 $K$.

Such an element is called completely free (or completely normal).

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]

References

[a1]	D. Blessenohl, K. Johnsen, "Eine Verschärfung des Satzes von der Normalbasis" J. Algebra , 103 (1986) pp. 141–159
[a2]	D. Hachenberger, "Finite fields: Normal bases and completely free elements" , Kluwer Acad. Publ. (1997)
[a3]	N. Jacobson, "Basic algebra" , I , Freeman (1985) (Edition: Second)
[a4]	R. Lidl, H. Niederreiter, "Finite fields" , Addison-Wesley (1983); second edition Cambridge University Press (1996) Zbl 0866.11069