Let $K$ be a number field and $a\in K^*$ of infinite order in $K^*$. How do I show that $$[K(\sqrt[n]{a},\zeta_n):K]\geq C\cdot n\cdot\varphi(n)$$ holds for all positive integers $n$, with a positive constant $C$ which only depends on $K$ and $a$?

Similar estimates often occur in connection with variations of Artin's primitive root conjecture. Hooley 1967 proved this for $K=\mathbb{Q}$ and Lenstra 1977 proved it for all $n$ which are coprime to some integer.

Usually one would approach the problem by Kummer theory and show that the set of $m$ such that $\sqrt[m]{a}$ is contained in some $K(\zeta_n)$ is finite. This, however, didn't work out yet.

I am thankful for any help.

Let $K$ be a number field and $a\in K^*$ of infinite order in $K^*$. How do I show that $$[K(\sqrt[n]{a},\zeta_n):K]\geq C\cdot n\cdot\varphi(n)$$ holds for all positive integers $n$, with a positive constant $C$ which only depends on $K$ and $a$?

Similar estimates often occur in connection with variations of Artin's primitive root conjecture. Hooley 1967 proved this for $K=\mathbb{Q}$ and Lenstra 1977 proved it for all $n$ which are coprime to some integer.

Usually one would approach the problem by Kummer theory and show that the set of $m$ such that $\sqrt[m]{a}$ is contained in some $K(\zeta_n)$ is finite. Thus, it suffices to show that for any prime $p$ the radical extension $K(\sqrt[p^e]{a})/K$ cannot be abelian for infinitely many $e$. This seems obvious since $a$ is not a root of unity, but I couldn't work out a proof yet.

I am thankful for any help.