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In [[mathematics]], a '''Galois extension''' is an [[Algebraic extension|algebraic field extension]] ''E''/''F'' that is [[normal extension|normal]] and [[separable extension|separable]]; or equivalently, ''E''/''F'' is [[algebraic extension|algebraic]], and the [[fixed field|field fixed]] by the [[automorphism]] group Aut(''E''/''F'') is precisely the base field ''F''. One says that such extension is '''Galois'''. The significance of being a Galois extension is that the extension has a [[Galois group]] and obeys the [[fundamental theorem of Galois theory]]. <ref> See the article [[Galois group]] for definitions of some of these terms and some examples.</ref>


A result of [[Emil Artin]] allows one to construct Galois extensions as follows: If ''E'' is a given field, and ''G'' is a finite group of automorphisms of ''E'', then ''E''/''F'' is a Galois extension, where ''F'' is the fixed field of ''G''.


==Characterization of Galois extensions==
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An important theorem of [[Emil Artin]] states that for a [[finite extension]] ''E''/''F'', each of the following statements is equivalent to the statement that ''E''/''F'' is Galois:
* ''E''/''F'' is a [[normal extension]] and a [[separable extension]].
* ''E'' is a [[splitting field]] of a [[separable polynomial]] with coefficients in ''F''.
* [''E'':''F''] = |Aut(''E''/''F'')|; that is, the [[degree (field theory)|degree]] of the field extension is equal to the [[order (group theory)|order]] of the automorphism group of ''E''/''F''.
 
==Examples==
 
[[Adjunction (field theory)|Adjoining]] to the [[rational number field]] the square root of 2 gives a Galois extension, while adjoining the cube root of 2 gives a non-Galois extension. Both these extensions are separable, because they have [[characteristic zero]]. The first of them is the splitting field of ''X''<sup>2</sup> &minus; 2; the second has [[Normal extension|normal closure]] that includes the complex [[cube roots of unity]], and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and ''X''<sup>3</sup> &minus; 2 has just one real root.
 
An [[algebraic closure]] <math>\bar K</math> of an arbitrary field <math>K</math> is Galois over <math>K</math> if and only if <math>K</math> is a [[perfect field]].
 
== See also ==
* {{cite book | author=Emil Artin | title=Galois Theory | publisher=Dover Publications | year=1998 | isbn=0-486-62342-4 | authorlink=Emil Artin}} ''(Reprinting of second revised edition of 1944, The University of Notre Dame Press)''.
* {{cite book | author=[[Jörg Bewersdorff]] | title=Galois Theory for Beginners: A Historical Perspective| publisher=American Mathematical Society | year=2006 | isbn=0-8218-3817-2}} .
* {{cite book|author=Harold M. Edwards | authorlink = Harold Edwards (mathematician)| title=Galois Theory | publisher=Springer-Verlag | year = 1984 | isbn=0-387-90980-X}} ''(Galois' original paper, with extensive background and commentary.)''
* {{cite journal| first= H. Gray | last=Funkhouser | title=A short account of the history of symmetric functions of roots of equations | journal=American Mathematical Monthly | year=1930 | volume= 37 | issue=7 | pages=357–365 | doi=10.2307/2299273| publisher= The American Mathematical Monthly, Vol. 37, No. 7| ref= harv| jstor= 2299273 }}
* {{springer|title=Galois theory|id=p/g043160}}
* {{cite book | author=Nathan Jacobson| title=Basic Algebra I (2nd ed) | publisher=W.H. Freeman and Company | year=1985 | isbn=0-7167-1480-9 | authorlink=Nathan Jacobson}} ''(Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)''
* {{Cite book | last1=Janelidze | first1=G. | last2=Borceux | first2=Francis | title=Galois theories | publisher=[[Cambridge University Press]] | isbn=978-0-521-80309-0 | year=2001 | ref=harv | postscript=<!--None-->}} (This book introduces the reader to the Galois theory  of [[Grothendieck]], and some generalisations, leading to Galois [[groupoids]].)
* {{Cite book | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebraic Number Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94225-4 | year=1994 | ref=harv | postscript=<!--None-->}}
* {{cite book|author=M. M. Postnikov | title=Foundations of Galois Theory | publisher=Dover Publications | year = 2004 | isbn=0-486-43518-0}}
* {{cite book|author=Joseph Rotman | title =Galois Theory (2nd edition)| publisher=Springer| year=1998 | isbn=0-387-98541-7}}
* {{Cite book | last1=Völklein | first1=Helmut | title=Groups as Galois groups: an introduction | publisher=[[Cambridge University Press]] | isbn=978-0-521-56280-5 | year=1996 | ref=harv | postscript=<!--None-->}}
* {{Cite book | last1=van der Waerden | first1=Bartel Leendert | author1-link=Bartel Leendert van der Waerden | title=Moderne Algebra |language= German |publisher=Springer | year=1931 | location=Berlin |ref=harv | postscript=<!--None-->}}.  '''English translation''' (of 2nd revised edition): {{Cite book | title = Modern algebra | publisher=Frederick Ungar |location= New York |year= 1949}} ''(Later republished in English by Springer under the title "Algebra".)'' 
* {{Cite web |title=(Some) New Trends in Galois Theory and Arithmetic |first=Florian |last=Pop |authorlink=Florian Pop|url=http://www.math.upenn.edu/~pop/Research/files-Res/Japan01.pdf |year=2001 |ref=harv |postscript=<!--None-->}}
 
== References ==
<references />
 
{{DEFAULTSORT:Galois Extension}}
[[Category:Galois theory]]
[[Category:Algebraic number theory]]
[[Category:Field extensions]]

Latest revision as of 18:47, 17 November 2014


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