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

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

Other equivalent statements are:

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

Examples

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

• Take any field ${\displaystyle E}$, any subgroup of ${\displaystyle \operatorname {Aut} (E)}$, and let ${\displaystyle F}$ be the fixed field.
• Take any field ${\displaystyle F}$, any separable polynomial in ${\displaystyle F[x]}$, and let ${\displaystyle E}$ 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 ${\displaystyle x^{2}-2}$; 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 ${\displaystyle x^{3}-2}$ has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.

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

References

1. See the article Galois group for definitions of some of these terms and some examples.

See also

• Artin, Emil (1998) [1944]. Galois Theory. Edited and with a supplemental chapter by Arthur N. Milgram. Mineola, NY: Dover Publications. ISBN 0-486-62342-4. MR 1616156.
• 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. MR 2251389.
• Edwards, Harold M. (1984). Galois Theory. Graduate Texts in Mathematics. 101. New York: Springer-Verlag. ISBN 0-387-90980-X. MR 0743418. (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. JSTOR 2299273.CS1 maint: ref=harv (link)
• "Galois theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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.CS1 maint: ref=harv (link) (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. MR 1282723.CS1 maint: ref=harv (link)
• 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. MR 2043554.
• Rotman, Joseph (1998). Galois Theory (Second ed.). Springer. doi:10.1007/978-1-4612-0617-0. ISBN 0-387-98541-7. MR 1645586.
• 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. MR 1405612.CS1 maint: ref=harv (link)
• van der Waerden, Bartel Leendert (1931). Moderne Algebra (in German). Berlin: Springer.CS1 maint: ref=harv (link). 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" (PDF).CS1 maint: ref=harv (link)