Let $K$ be any field containing all the $n^{\text{th}}$ roots of unity. Let $A = \{\xi_n\}_n$ be a collection of primitive roots of unity (so each $\xi_n$ is a primitive $n^{\text{th}}$ roots of unity). It is easy to prove that, for each $n,m$, the element $\xi_{n·m}^n$ is again a primitive $m^{\text{th}}$ root of unity, which $\textit{might not be}$ the primitive one $\xi_m$ which was already chosen. One can change $\xi_m$ by the new one $\xi_{n·m}^n$, but by doing so it could happen that other roots are not compatible anymore, for example both $\xi_{2·n}^2$ and $\xi_{3·n}^3$ are primitive $n^{\text{th}}$ roots of unity, which $\textit{may not be}$ the same.
My question is: is it possible to find a $\textit{compatible}$ collection $\{\eta_n\}_n$ of primitive roots of unity, meaning that the equation $$\eta_{n·m}^n = \eta_m$$ holds for all $n,m$? It is easily verified that inside the complex numbers such collection can easily be chosen, taking for example $\eta_n = e^{\frac{2 \pi i}{n}}$.
I do not want to stick with the complex numbers case because my aim is to prove it abstractly, which I think it could be possible.
Thanks in advance!