Editing Galois extension

{{Short description|Algebraic field extension}}
In [[mathematics]], a '''Galois extension''' is an [[Algebraic extension|algebraic]] [[field extension]] ''E''/''F'' that is [[normal extension|normal]] and [[separable extension|separable]];{{sfn|Lang|2002|p=262}} or equivalently, ''E''/''F'' is algebraic, and the [[Fixed field|field fixed]] by the [[automorphism group]] Aut(''E''/''F'') is precisely the base [[Field (mathematics)|field]] ''F''. The significance of being a Galois extension is that the extension has a [[Galois group]] and obeys the [[fundamental theorem of Galois theory]].{{efn|See the article [[Galois group]] for definitions of some of these terms and some examples.}}

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'' with fixed field ''F'', then ''E''/''F'' is a Galois extension.{{sfn|Lang|2002|p=264|loc=Theorem 1.8}}

The property of an extension being Galois behaves well with respect to [[Composite field (mathematics)| field composition and intersection]].{{sfn|Milne|2022|p=40f|loc=ch. 3 and 7}}

==Characterization of Galois extensions==
An important theorem of [[Emil Artin]] states that for a [[finite extension]] <math>E/F,</math> each of the following statements is equivalent to the statement that <math>E/F</math> is Galois:

*<math>E/F</math> is a [[normal extension]] and a [[separable extension]].
*<math>E</math> is a [[splitting field]] of a [[separable polynomial]] with coefficients in <math>F.</math>
*<math>|\!\operatorname{Aut}(E/F)| = [E:F],</math> that is, the number of automorphisms equals the [[degree (field theory)|degree]] of the extension.

Other equivalent statements are:

*Every irreducible polynomial in <math>F[x]</math> with at least one root in <math>E</math> splits over <math>E</math> and is separable.
*<math>|\!\operatorname{Aut}(E/F)| \geq [E:F],</math> that is, the number of automorphisms is at least the degree of the extension.
*<math>F</math> is the fixed field of a subgroup of <math>\operatorname{Aut}(E).</math>
*<math>F</math> is the fixed field of <math>\operatorname{Aut}(E/F).</math>
*There is a one-to-one [[Fundamental theorem of Galois theory#Explicit description of the correspondence|correspondence]] between subfields of <math>E/F</math> and subgroups of <math>\operatorname{Aut}(E/F).</math>

An infinite field extension <math>E/F</math> is Galois if and only if <math>E</math> is the union of finite Galois subextensions <math>E_i/F</math> indexed by an (infinite) index set <math>I</math>, i.e. <math>E=\bigcup_{i\in I}E_i</math> and the Galois group is an [[inverse limit]] <math>\operatorname{Aut}(E/F)=\varprojlim_{i\in I}{\operatorname{Aut}(E_i/F)}</math> where the inverse system is ordered by field inclusion  <math>E_i\subset E_j</math>.{{sfn|Milne|2022|p=102|loc=example 7.26}}

==Examples==
There are two basic ways to construct examples of Galois extensions.

* Take any field <math>E</math>, any finite subgroup of <math>\operatorname{Aut}(E)</math>, and let <math>F</math> be the fixed field.
* Take any field <math>F</math>, any separable polynomial in <math>F[x]</math>, and let <math>E</math> be its [[splitting field]].

[[Adjunction (field theory)|Adjoining]] to the [[rational number field]] the [[square root of 2]] gives a Galois extension, while adjoining the cubic 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 <math>x^2 -2</math>; the second has [[Normal extension|normal closure]] that includes the complex [[Root_of_unity | cubic 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 <math>x^3 -2</math> has just one real root. For more detailed examples, see the page on the [[fundamental theorem of Galois theory]].

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]].

== Notes ==
{{Notelist}}

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
*{{Lang Algebra|3rd}}
{{refend}}

== Further reading ==
{{refbegin|2}}
*{{cite book|last=Artin | first=Emil | title=Galois Theory | publisher=Dover Publications | year=1998 | orig-year=1944 | isbn=0-486-62342-4 | authorlink=Emil Artin | mr=1616156 | location=Mineola, NY | others=Edited and with a supplemental chapter by Arthur N. Milgram}}
*{{cite book|first=Jörg | last=Bewersdorff | authorlink=Jörg Bewersdorff|title=Galois theory for beginners | others=Translated from the second German (2004) edition by David Kramer | publisher=American Mathematical Society | year=2006 | isbn=0-8218-3817-2 | mr=2251389 | series=Student Mathematical Library | volume=35|doi=10.1090/stml/035| s2cid=118256821 }}
*{{cite book | first=Harold M. | last=Edwards | authorlink=Harold Edwards (mathematician) | title=Galois Theory | publisher=Springer-Verlag | location=New York | year=1984 | isbn=0-387-90980-X | mr=0743418 | series=[[Graduate Texts in Mathematics]] | volume=101 | url-access=registration | url=https://archive.org/details/galoistheory00edwa_0 }} ''(Galois' original paper, with extensive background and commentary.)''
*{{cite journal|first= H. Gray | last=Funkhouser | authorlink = Howard G. 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| jstor= 2299273 }}
*{{springer|title=Galois theory|id=p/g043160}}
* {{cite book| first=Nathan | last=Jacobson| title=Basic Algebra I | edition=2nd | 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 }} (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 | mr=1282723 | series=Graduate Texts in Mathematics | volume=110 | edition=Second | doi= 10.1007/978-1-4612-0853-2}}
*{{cite book|first=Mikhail Mikhaĭlovich | last=Postnikov | title=Foundations of Galois Theory | others=With a foreword by P. J. Hilton. Reprint of the 1962 edition. Translated from the 1960 Russian original by Ann Swinfen | publisher=Dover Publications | year = 2004 | isbn=0-486-43518-0 | mr=2043554}}
*{{cite book |last=Milne |first=James S. |date=2022 |title=Fields and Galois Theory (v5.10) |url=https://www.jmilne.org/math/CourseNotes/ft.html}}
*{{cite book|first=Joseph | last=Rotman | title =Galois Theory | series=Universitext | edition=Second | publisher=Springer| year=1998 | isbn=0-387-98541-7 | mr=1645586 | doi=10.1007/978-1-4612-0617-0}}
*{{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 | series=Cambridge Studies in Advanced Mathematics | volume=53 | mr=1405612 | doi=10.1017/CBO9780511471117 | url-access=registration | url=https://archive.org/details/groupsasgaloisgr0000volk }}
*{{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 }}.   '''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 }}
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{{DEFAULTSORT:Galois Extension}}
[[Category:Galois theory]]
[[Category:Algebraic number theory]]
[[Category:Field extensions]]