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Difference between revisions of "Normal basis theorem"

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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 $K$.
 
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]]].
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Such an element is called completely free (or completely normal).  
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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]]]. 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 [[#References|[a4]]]
  
 
====References====
 
====References====
 
<table>
 
<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>
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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>
<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">[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>
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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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<TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Lidl,  H. Niederreiter, "Finite fields" , Addison-Wesley  (1983)  {{ZBL|0554.12010}}; second edition Cambridge University Press (1996)  {{ISBN|0-521-39231-4}}  {{ZBL|0866.11069}}</TD></TR>
 
</table>
 
</table>
  
 
{{TEX|done}}
 
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[[Category:Field theory and polynomials]]

Latest revision as of 19:12, 24 November 2023

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) Zbl 0554.12010; second edition Cambridge University Press (1996) ISBN 0-521-39231-4 Zbl 0866.11069
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