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 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.
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\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 F[x] with at least one root in E splits over E and is separable.
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\operatorname{Aut}(E/F)| \geq [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 \operatorname{Aut}(E).
F is the fixed field of \operatorname{Aut}(E/F).
There is a one-to-one correspondence between subfields of E/F and subgroups of \operatorname{Aut}(E/F).
Examples
There are two basic ways to construct examples of Galois extensions.
Take any field E, any subgroup of \operatorname{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 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 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 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 \bar K of an arbitrary field K is Galois over K if and only if K is a perfect field.
Notes
Citations
References
Further reading
(Galois' original paper, with extensive background and commentary.)
(Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)
(This book introduces the reader to the Galois theory  of Grothendieck, and some generalisations, leading to Galois groupoids.)
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English translation (of 2nd revised edition):  (Later republished in English by Springer under the title "Algebra".)
