Editing Galois extension (section)

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

* Take any field <math>E</math>, any finite subgroup of <math>\operatorname{Aut}(E)</math>, and let <math>F</math> be the fixed field.
* Take any field <math>F</math>, any separable polynomial in <math>F[x]</math>, and let <math>E</math> be its [[splitting field]].

[[Adjunction (field theory)|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 <math>x^2 -2</math>; the second has [[Normal extension|normal closure]] that includes the complex [[Root_of_unity | 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 <math>x^3 -2</math> has just one real root. For more detailed examples, see the page on the [[fundamental theorem of Galois theory]].

An [[algebraic closure]] <math>\bar K</math> of an arbitrary field <math>K</math> is Galois over <math>K</math> if and only if <math>K</math> is a [[perfect field]].