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. One says that such extension is Galois. 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 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.
• [E:F] = |Aut(E/F)|; that is, the degree of the field extension is equal to the order of the automorphism group of E/F.

Examples

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 X² − 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 X³ − 2 has just one real root.

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

See also

• Emil Artin (1998). Galois Theory. Dover Publications. ISBN 0-486-62342-4. (Reprinting of second revised edition of 1944, The University of Notre Dame Press).
• Jörg Bewersdorff (2006). Galois Theory for Beginners: A Historical Perspective. American Mathematical Society. ISBN 0-8218-3817-2. .
• Harold M. Edwards (1984). Galois Theory. Springer-Verlag. ISBN 0-387-90980-X. (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. 37 (7). The American Mathematical Monthly, Vol. 37, No. 7: 357–365. doi:10.2307/2299273. JSTOR 2299273. {{cite journal}}: Invalid |ref=harv (help)
• "Galois theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Nathan Jacobson (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. {{cite book}}: Invalid |ref=harv (help) (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.)
• Lang, Serge (1994). Algebraic Number Theory. Berlin, New York: Springer-Verlag. ISBN 978-0-387-94225-4. {{cite book}}: Invalid |ref=harv (help)
• M. M. Postnikov (2004). Foundations of Galois Theory. Dover Publications. ISBN 0-486-43518-0.
• Joseph Rotman (1998). Galois Theory (2nd edition). Springer. ISBN 0-387-98541-7.
• Völklein, Helmut (1996). Groups as Galois groups: an introduction. Cambridge University Press. ISBN 978-0-521-56280-5. {{cite book}}: Invalid |ref=harv (help)
• van der Waerden, Bartel Leendert (1931). Moderne Algebra (in German). Berlin: Springer. {{cite book}}: Invalid |ref=harv (help). 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). {{cite web}}: Invalid |ref=harv (help)