View source for 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., [[#References|[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 [[#References|[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 [[#References|[a2]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Blessenohl,   K. Johnsen,   "Eine Verschärfung des Satzes von der Normalbasis"  ''J. Algebra'' , '''103'''  (1986)  pp. 141–159</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Hachenberger,   "Finite fields: Normal bases and completely free elements" , Kluwer Acad. Publ.  (1997)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Jacobson,   "Basic algebra" , '''I''' , Freeman  (1985)  (Edition: Second)</TD></TR>
</table>

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