Compute the splitting field of over and its degree.
Note that factors as . (Using only the difference of squares and sum of cubes formulas familiar to middle schoolers.) Using the quadratic formula (again familiar to middle schoolers) we see that the roots of are and .
Now if is a 6th root of 1, then is a root of , where denotes the positive real 6th root of 4 (aka the positive cube root of 2). (Middle schoolers could verify that.) Evidently, then, the splitting field of is .
Now has degree 3 over , and has degree 2 over . (Use Eisenstein for both.) By Corollary 22 on page 529 of D&F, then, has degree (as a middle schooler could compute) over .
So we have proved not only that the splitting field of has degree 6 over , but, in a metamathematical twist, also that a middle schooler could prove this as well.