The full question is Let $K$ over$F$ be a Galois extension whose Galois group is $S_3$. Prove that K does not contain a cyclic extension of degree 3.
Is this because $S_3$ does not have a subgroup of index 3? I'm not sure what to do here. Any suggestions would be great
Not quite. Let $K/F$ be Galois with Galois group, $G$. Consider an intermediate extension $F\subseteq M\subseteq K$. When we say that the extension $M/F$ is cyclic, we mean that it is Galois with cyclic Galois group. The Galois part is important, since the Galois intermediate extensions $M/F$ are precisely those whose corresponding subgroup $N\subseteq G$ is normal. Moreover, $\operatorname{Gal}(M/F)\cong G/N$. Together these imply that $K/F$ contains a cyclic intermediate extension of degree 3 if and only if $G$ contains a normal subgroup of index 3. $S_3$ has subgroups of index 3 (those generated by a single 2-cycle), but all of these subgroups are conjugate, and hence none are normal. Therefore, while there are degree 3 intermediate extensions, there are no intermediate extensions that are cyclic of degree 3.