Show that the splitting field of is a simple extension of .
Let be a primitive 8th root of unity and note that is a root of . (Make sure you understand why this is true.) Further, as and , note that are all roots of as well. Thus, we have found the four roots of this quartic polynomial, so the splitting field of is . As is a field, all the integral powers of must be elements of it, so we may write , showing that is a simple extension of .