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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n1300901.png" /> be a (finite-dimensional) [[Galois extension|Galois extension]] of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n1300902.png" />. Then there exists a normal basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n1300903.png" />, that is, a basis consisting of an orbit of the [[Galois group|Galois group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n1300904.png" />. Thus, an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n1300905.png" /> generates a normal basis if and only if its conjugates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n1300906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n1300907.png" />, are linearly independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n1300908.png" />; see, e.g., [[#References|[a3]]]. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n1300909.png" /> is called a normal basis generator or a free element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n13009010.png" />. A far-reaching strengthening of the normal basis theorem is due to D. Blessenohl and K. Johnsen [[#References|[a1]]]: There exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n13009011.png" /> that is simultaneously free in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n13009012.png" /> for every intermediate field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n13009013.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130090/n13009014.png" /> is a [[Galois field|Galois field]], a constructive treatment of normal bases and completely free elements can be found in [[#References|[a2]]].
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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====
 
====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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<table>
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<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>
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<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>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Jacobson,  "Basic algebra" , '''I''' , Freeman  (1985)  (Edition: Second)</TD></TR>
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</table>
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{{TEX|done}}

Revision as of 18:26, 2 November 2014

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].

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)
How to Cite This Entry:
Normal basis theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_basis_theorem&oldid=34236
This article was adapted from an original article by Dieter Jungnickel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article