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. 
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 formula_1 each of the following statements is equivalent to the statement that formula_2 is Galois:
Other equivalent statements are:
Examples.
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 formula_23; 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 formula_24 has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.
An algebraic closure formula_25 of an arbitrary field formula_26 is Galois over formula_26 if and only if formula_26 is a perfect field.