Prove that E=Q(2^(1/3)) is not a subfield of any cyclotomic field over Q.
First I note that the extension E/Q is not Galois. It has only trivial automorphism group and is an extension of degree 3.
The Galois closure is F = Q(2^(1/3),z_3), where z_3 is a third root of unity.
I try to proceed by contradiction.
Suppose E is contained in Q(z_n) some cyclotomic field.
The Galois closure of E, F, must be contained in Q(z_n). So z_3 must be in z_n.
At this point I am stuck. I tried to show that 3|n. I know that n does not equal 3. Also if 3|n then n is not prime.
I can't see how to proceed (or if this is even a good method)