Galois extension

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 $F.$
• $|\backslash !\backslash operatorname\{Aut\}(E/F)|\; =\; [E:F],$ that is, the number of automorphisms equals the degree of the extension.

Other equivalent statements are:

• Every irreducible polynomial in $F[x]$ with at least one root in $E$ splits over $E$ and is separable.
• $|\backslash !\backslash operatorname\{Aut\}(E/F)|\; \backslash geq\; [E:F],$ that is, the number of automorphisms is at least the degree of the extension.
• $F$ is the fixed field of a subgroup of $\backslash operatorname\{Aut\}(E).$
• $F$ is the fixed field of $\backslash operatorname\{Aut\}(E/F).$
• There is a one-to-one correspondence between subfields of $E/F$ and subgroups of $\backslash operatorname\{Aut\}(E/F).$

An infinite field extension $E/F$ is Galois if and only if $E$ is the union of finite Galois subextensions $E\_i/F$ indexed by an (infinite) index set $I$, i.e. $E=\backslash bigcup\_\{i\backslash in\; I\}E\_i$ and the Galois group is an inverse limit $\backslash operatorname\{Aut\}(E/F)=\backslash varprojlim\_\{i\backslash in\; I\}\{\backslash operatorname\{Aut\}(E\_i/F)\}$ where the inverse system is ordered by field inclusion $E\_i\backslash subset\; E\_j$.{{sfn|Milne|2022|p=102|loc=example 7.26}}

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

• Take any field $E$, any finite subgroup of $\backslash operatorname\{Aut\}(E)$, and let $F$ be the fixed field.
• Take any field $F$, any separable polynomial in $F[x]$, and let $E$ be its splitting field.

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 $x^2\; -2$; the second has normal closure that includes the complex 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 $x^3\; -2$ has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.

An algebraic closure $\backslash bar\; K$ of an arbitrary field $K$ is Galois over $K$ if and only if $K$ is a perfect field.

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• {{Lang Algebra|3rd}}

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• {{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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Category:Galois theory

Category:Algebraic number theory

Category:Field extensions