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Difference between revisions of "Normal extension"

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''of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675001.png" />''
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''of a field $K$''
  
An algebraic field extension (cf. [[Extension of a field|Extension of a field]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675002.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675003.png" /> satisfying one of the following equivalent conditions:
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An algebraic field extension (cf. [[Extension of a field]]) $L$ of $K$ satisfying one of the following equivalent conditions:
  
1) any imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675004.png" /> in the [[Algebraic closure|algebraic closure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675005.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675006.png" /> comes from an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675007.png" />;
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1) any imbedding of $L$ in the [[algebraic closure]] $\bar K$ of $K$ comes from an automorphism of $L$;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675008.png" /> is the splitting field of some family of polynomials with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675009.png" /> (cf. [[Splitting field of a polynomial|Splitting field of a polynomial]]);
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2) $L$ is the splitting field of some family of polynomials with coefficients in $K$ (cf. [[Splitting field of a polynomial]]);
  
3) any polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750010.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750011.png" />, irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750012.png" /> and having a root in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750013.png" />, splits in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750014.png" /> into linear factors.
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3) any polynomial $f(x)$ with coefficients in $K$, irreducible over $K$ and having a root in $L$, splits in $L$ into linear factors.
  
For every algebraic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750015.png" /> there is a maximal intermediate subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750016.png" /> that is normal over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750017.png" />; this is the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750019.png" /> ranges over all imbeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750021.png" />. There is also a unique minimal normal extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750022.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750023.png" />. This is the composite of all fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750024.png" />. It is called the normal closure of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750025.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750028.png" /> are normal extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750029.png" />, then so are the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750030.png" /> and the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750031.png" />. However, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750033.png" /> are normal extensions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750034.png" /> need not be normal.
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For every [[algebraic extension]] $F/K$ there is a maximal intermediate subfield $L$ that is normal over $K$; this is the field $L = \bigcap_\sigma F^\sigma$, where $\sigma$ ranges over all imbeddings of $F$ in $\bar K$. There is also a unique minimal normal extension of $K$ containing $F$. This is the composite of all fields $F^\sigma$. It is called the ''normal closure'' of the field $F$ relative to $K$. If $L_1$ and $L_2$ are normal extensions of $K$, then so are the intersection $L_1 \cap L_2$ and the composite $L_1 \cdot L_2$. However, when $L/K'$ and $K'/K$ are normal extensions, $L/K$ need not be normal.
  
For fields of characteristic zero every normal extension is a Galois extension. In general, a normal extension is a [[Galois extension|Galois extension]] if and only if it is separable (cf. [[Separable extension|Separable extension]]).
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For fields of characteristic zero every normal extension is a Galois extension. In general, a normal extension is a [[Galois extension]] if and only if it is a [[separable extension]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Postnikov,  "Fundamentals of Galois theory" , Noordhoff  (1962)  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Postnikov,  "Fundamentals of Galois theory" , Noordhoff  (1962)  (Translated from Russian)</TD></TR>
 +
</table>
 +
 
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{{TEX|done}}

Latest revision as of 18:30, 11 April 2016

of a field $K$

An algebraic field extension (cf. Extension of a field) $L$ of $K$ satisfying one of the following equivalent conditions:

1) any imbedding of $L$ in the algebraic closure $\bar K$ of $K$ comes from an automorphism of $L$;

2) $L$ is the splitting field of some family of polynomials with coefficients in $K$ (cf. Splitting field of a polynomial);

3) any polynomial $f(x)$ with coefficients in $K$, irreducible over $K$ and having a root in $L$, splits in $L$ into linear factors.

For every algebraic extension $F/K$ there is a maximal intermediate subfield $L$ that is normal over $K$; this is the field $L = \bigcap_\sigma F^\sigma$, where $\sigma$ ranges over all imbeddings of $F$ in $\bar K$. There is also a unique minimal normal extension of $K$ containing $F$. This is the composite of all fields $F^\sigma$. It is called the normal closure of the field $F$ relative to $K$. If $L_1$ and $L_2$ are normal extensions of $K$, then so are the intersection $L_1 \cap L_2$ and the composite $L_1 \cdot L_2$. However, when $L/K'$ and $K'/K$ are normal extensions, $L/K$ need not be normal.

For fields of characteristic zero every normal extension is a Galois extension. In general, a normal extension is a Galois extension if and only if it is a separable extension.

References

[1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2] S. Lang, "Algebra" , Addison-Wesley (1984)
[3] M.M. Postnikov, "Fundamentals of Galois theory" , Noordhoff (1962) (Translated from Russian)
How to Cite This Entry:
Normal extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_extension&oldid=15251
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article