# The direct limit of roots of unity

Let $$\mu_n$$ denote the group of $$n$$-th roots of unity. In $$\mathbb{C}$$, this group has exactly $$n$$ elements. For a positive integer $$n$$ and a prime number $$p$$, using the canonical isomorphisms $$\mu_{p^n} \cong \mathbb{Z}/p^n\mathbb{Z}$$ for all $$n$$ as a direct system of factor groups, and the multiplication-by-$$p$$ homomorphisms $$\mathbb{Z}/p^n\mathbb{Z} \rightarrow \mathbb{Z}/p^{n+1}\mathbb{Z},$$ one sees that the direct limit of this system is the Prüfer group $$\mathbb{Z}(p^\infty) = \{x \in \mathbb{C}^\times: x^{p^n} = 1\,\, \mathrm{for\,\,some} \,\,n\}.$$

Similarly, as seen in this question Union of all finite cyclic groups, we can consider a direct system of cyclic group $$\mathbb{Z}/n\mathbb{Z}$$ and homomorphisms $$\mathbb{Z}/n\mathbb{Z} \rightarrow \mathbb{Z}/m\mathbb{Z}$$ if $$n|m$$. The resulting direct limit is seen to be $$\mathbb{Q}/\mathbb{Z}$$, the set of rational numbers under addition modulo $$1$$. We have $$\mathbb{Q}/\mathbb{Z} = \bigoplus_p \mathbb{Z}(p^\infty).$$

Question. Can we view $$\mathbb{Q}/\mathbb{Z}$$ as being isomorphic to $$\mu_\infty = \{x \in \mathbb{C}^\times:x^n = 1\,\,\mathrm{for\,\,some}\,\, n\in \mathbb{Z}^+\}$$, i.e., the subgroup of all roots of unity? If so, is there any way to define the isomorphism?

Yes, send $$a/b \in \mathbb{Q}/\mathbb{Z}$$ to $$\zeta_{b}^{a}$$, where $$\zeta_{b}$$ is a primitive $$b$$-th root of unity.
• This is $\frac ab\mapsto e^{2\pi i\frac ab}$.
• This is ill- or under-defined. You need to make an infinite number of arbitrary choices - a different primitive root $\zeta_b$ for each $b$ - and you need to synchronize the choices appropriately (so for example the primitive $9$th root cubed equals the primitive $3$rd root, etc.) Way simpler to just say $x\mapsto\exp(2\pi i x)$.