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 of an arbitrary field is Galois over if and only if is a perfect field.
Read more about this topic: Galois Extension
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