# Galois extension – Wikipedia

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]}

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[edit]

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:

Other equivalent statements are:

## Examples[edit]

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

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.

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

## Citations[edit]

## References[edit]

## Further reading[edit]

- 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. Vol. 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. Vol. 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. - “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. (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. Vol. 110 (Second ed.). Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4612-0853-2. ISBN 978-0-387-94225-4. MR 1282723. - 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*. Universitext (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. Vol. 53. Cambridge University Press. doi:10.1017/CBO9780511471117. ISBN 978-0-521-56280-5. MR 1405612. - van der Waerden, Bartel Leendert (1931).
*Moderne Algebra*(in German). 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” (PDF).

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