Galois extension

Short description: Algebraic field extension

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.

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 [math]\displaystyle{ E/F, }[/math] each of the following statements is equivalent to the statement that [math]\displaystyle{ E/F }[/math] is Galois:

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

Other equivalent statements are:

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

Examples

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

• Take any field [math]\displaystyle{ E }[/math], any finite subgroup of [math]\displaystyle{ \operatorname{Aut}(E) }[/math], and let [math]\displaystyle{ F }[/math] be the fixed field.
• Take any field [math]\displaystyle{ F }[/math], any separable polynomial in [math]\displaystyle{ F[x] }[/math], and let [math]\displaystyle{ E }[/math] 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 [math]\displaystyle{ x^2 -2 }[/math]; 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 [math]\displaystyle{ 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]\displaystyle{ \bar K }[/math] of an arbitrary field [math]\displaystyle{ K }[/math] is Galois over [math]\displaystyle{ K }[/math] if and only if [math]\displaystyle{ K }[/math] is a perfect field.

References

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4 

Further reading

• Artin, Emil (1998). Galois Theory. Edited and with a supplemental chapter by Arthur N. Milgram. Mineola, NY: Dover Publications. ISBN 0-486-62342-4. 
• Bewersdorff, Jörg (2006). Galois theory for beginners. Student Mathematical Library. 35. Translated from the second German (2004) edition by David Kramer. American Mathematical Society. doi:10.1090/stml/035. ISBN 0-8218-3817-2. 
• Edwards, Harold M. (1984). Galois Theory. Graduate Texts in Mathematics. 101. New York: Springer-Verlag. ISBN 0-387-90980-X. https://archive.org/details/galoistheory00edwa_0.  (Galois' original paper, with extensive background and commentary.)
• Funkhouser, H. Gray (1930). "A short account of the history of symmetric functions of roots of equations". American Mathematical Monthly (The American Mathematical Monthly, Vol. 37, No. 7) 37 (7): 357–365. doi:10.2307/2299273. 
• Hazewinkel, Michiel, ed. (2001), "Galois theory", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/g043160 
• Jacobson, Nathan (1985). Basic Algebra I (2nd ed.). W.H. Freeman and Company. ISBN 0-7167-1480-9.  (Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)
• Janelidze, G.; Borceux, Francis (2001). Galois theories. Cambridge University Press. ISBN 978-0-521-80309-0.  (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.)
• Lang, Serge (1994). Algebraic Number Theory. Graduate Texts in Mathematics. 110 (Second ed.). Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4612-0853-2. ISBN 978-0-387-94225-4. 
• Postnikov, Mikhail Mikhaĭlovich (2004). Foundations of Galois Theory. With a foreword by P. J. Hilton. Reprint of the 1962 edition. Translated from the 1960 Russian original by Ann Swinfen. Dover Publications. ISBN 0-486-43518-0. 
• Rotman, Joseph (1998). Galois Theory. Universitext (Second ed.). Springer. doi:10.1007/978-1-4612-0617-0. ISBN 0-387-98541-7. 
• Völklein, Helmut (1996). Groups as Galois groups: an introduction. Cambridge Studies in Advanced Mathematics. 53. Cambridge University Press. doi:10.1017/CBO9780511471117. ISBN 978-0-521-56280-5. https://archive.org/details/groupsasgaloisgr0000volk. 
• van der Waerden, Bartel Leendert (1931) (in German). Moderne Algebra. Berlin: Springer. . English translation (of 2nd revised edition): Modern algebra. New York: Frederick Ungar. 1949.  (Later republished in English by Springer under the title "Algebra".)
• Pop, Florian (2001). "(Some) New Trends in Galois Theory and Arithmetic". http://www.math.upenn.edu/~pop/Research/files-Res/Japan01.pdf.