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.