When do we use the $n^{th}$ primitive root of unity, $\zeta_{n}$, when trying to find roots of a polynomial?
For example, let $f=x^{3}-2\in \mathbb{Q}$[x]. The roots of $f$, are $\sqrt[3]{2},~\zeta_{3}\sqrt[3]{2},~\zeta_{3}^{2}\sqrt[3]{2}$. I would have only guessed $\sqrt[3]{2}$.
In general, given a field $\mathbb{F}$ and $f\in\mathbb{F}$[x], where, $f=x^{n}-a$, why do we sometimes use the primitive roots of unity?