Existence of a compatible system of roots of unity

A compatible system of roots of unity exists: there is a function $n\mapsto\zeta_n\in\mathbb C$ such that

1. for every square-free positive integer $n$, $\zeta_n$ is a primitive $n$-th root of unity, and
2. $\zeta_{mn}=\zeta_m\zeta_n$ for all coprime square-free positive integers $m,n$.

This is the "pick" step of the source's construction: it shows the hypotheses of the norm and congruence relations can be met, so those statements are not vacuous.