prove that the Galois group $Gal(L:K)$ is cyclic Let $L$ be a field extension of a field $K$. Suppose $L=K(a)$, and $a^n\in K$ for some integer $n$. If $L$ is Galois extension of $K$ (i.e., $L$ is the splitting field of $f(x)\in K[x]$, and $f$ is separable over $L$), then prove that the Galois group $Gal(L:K)$ is cyclic. 

Problem is true if we impose the condition $K$ contains a primitive root of unity.
Proof: Letting $G_n$ be group of all root unity we can show there is an injection $Gal(E/K) \to G_n$ given by $\sigma \to \frac{\sigma(a)}{a}$. So, we done.
If we impose the condition that $a^{i}$ doesn't belong to $K$ for all $i<n$ the above map would be isomorphism $\implies$ galois group has order $n$
