Let $K$ be a field with $\operatorname{char}(K) \neq 2$. Further, let $f = X^4-a \in K[X]$ be irreducible and $L$ be its splitting. Then $\operatorname{Gal}(L/K)$ is not cyclic.
I tried to find a counterexample to this a while now and didn't find any so I suspect the statement is true. However I am stuck with proving it:
$f$ is separable since $\operatorname{char}(K) \neq 2$ hence $\operatorname{Gal}(L/K) = G$ is isomorphic to a subgroup of $S_4$ and the order of $G$ divides $24$. Further, for every pair $1 \leq i,j \leq 4$ there exists a $\sigma \in G$ s.t $\sigma(\alpha_i) = \alpha_j$, where $\alpha_i,\alpha_j$ are two roots of $f$. How do I proceed from here on? I tried a proof by contradiction but didn't get very far with it. Hints or advice is greatly appreciated!