Primitive radical extension with $n$-th root with Galois group smaller than $n$

Question

It is a well-known theorem in Field Theory that if $F$ is a field:

1. Which contains the $n$-th roots of unity for some $n\ge 1$.

2. Is of characteristic not dividing $n$.

   And if $0\ne a\in F$ is some element, then the primitive radical extension $F(\sqrt[n]{a})/F$ is a Galois extension with $\textbf{Gal}(F(\sqrt[n]{a})/F)\cong \mathbb{Z}_d$ such that $d|n$.

I am looking for examples (preferably over characteristic 0, is such examples exist) for such extensions where $d<n$.

Answer 1

Let $F$ be the extension of the rationals got by joining a primitive 15th root of unity. Take $a=8$. Then the extension $F[ 8^{1/15}]$ of $F$ will have Galois group cyclic of order 5. (Because 8 already has cubic roots in $F$).