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 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.
- |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.
- |Aut(E/F)| ≥ [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 Aut(E).
- F is the fixed field of Aut(E/F).
- There is a one-to-one correspondence between subfields of E/F and subgroups of Aut(E/F).
Examples
There are two basic ways to construct examples of Galois extensions.
- Take any field E, any subgroup of 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 cube 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 cube 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 of an arbitrary field is Galois over if and only if is a perfect field.
References
- ^ See the article Galois group for definitions of some of these terms and some examples.
See also
- Artin, Emil (1998). 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.
- Hazewinkel, Michiel, ed. (2001) [1994], "Galois theory", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- 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. 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 (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.
- 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).