subfield maximal w.r.t. $\sqrt[3]{2}$ not in it. Let $K$ be a field maximal with respect to the properties that $K\leq\overline{\mathbb{Q}}$ and $\sqrt[3]{2}\not\in K$(which exists by Zorn's Lemma). I was asked to show that if $L/K$ is an algebraic extension and cyclic, then $[L:K]$ is either a power of 2 or a power of 3. Any ideas?

Let $G=Gal(L/K)$ and $p=[K(\sqrt[3]2):K]$, then $p=2$, or, $p=3$.
Let $q$ be a prime that divides $n=|G|$, and $H\leq G$ a subgroup of index $q$, the group $H$ exists because $G$ is cyclic. Let $F$ be the fixed field of $H$. Then, 
$$[F:K]=[G:H]=q,$$ 
but $\sqrt[3]2\in F$, then $[F:K]=[F:K(\sqrt[3]2)][K(\sqrt[3]2):K]$, hence $p|q\Rightarrow p=q$. We conclude $n=2^r$, or, $n=3^r$ as desired.
