- Characterization of Galois extensions
- Examples
- References
- See also
In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base 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. [1]
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.
Characterization of Galois extensions
An important theorem of Emil Artin states that for a finite extension 
, each of the following statements is equivalent to the statement that is Galois:
- is a normal extension and a separable extension.
-
is a splitting field of a separable polynomial with coefficients in 
. -
= 
, that is, the number of automorphisms equals the degree of the extension.
Other equivalent statements are:
- Every irreducible polynomial in
with at least one root in splits over and is separable. -
≥ , that is, the number of automorphisms is at least the degree of the extension. - is the fixed field of a subgroup of
. - is the fixed field of .
- There is a one-to-one correspondence between subfields of and subgroups of .
Examples
There are two basic ways to construct examples of Galois extensions.
- Take any field , any subgroup of , and let be the fixed field.
- Take any field , any separable polynomial in , and let 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 
; 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 
has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.
An algebraic closure 
of an arbitrary field 
is Galois over if and only if is a perfect field.
References
See also
- {{cite book | last=Artin | first=Emil | title=Galois Theory | publisher=Dover Publications | year=1998 | 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}}
- {{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}} (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| ref= harv| 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 | ref=harv | postscript=}} (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 | 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|first=Joseph | last=Rotman | title =Galois Theory | 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 | ref=harv | series=Cambridge Studies in Advanced Mathematics | volume=53 | mr=1405612 | doi=10.1017/CBO9780511471117}}
- {{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=}}. 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=}}
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