Chapter 17, Example 8 says that f(x) = x^5 + 2x + 4 is irreducible over Q. This means that F = Q[x]/<x^5 + 2x + 4> is a field. If beta is a zero of f(x), then Q(beta) has degree 5 over Q. So, any element in Q(beta) has a degree that divides 5.
[Q(2^(1/2)):Q] = 2, [Q(2^(1/3)):Q] = 3, [Q(2^(1/4)):Q] = 4
Since 2, 3, and 4 do not divide 5, 2^(1/2), 2^(1/3), and 2^(1/4) do not belong to Q(beta).
