Unit roots group is isomorphic to $\Bbb{Q}/\Bbb{Z}\left[\frac{1}{p}\right]$ in a field of characteristic $p\ge0$

Question

Let $K$ be a field so that the group of all unit roots of all orders $\mu_\infty=\bigcup_n {\mu_n}$ (where $\mu_n=\{x\in K\mid x^n=1\}$) splits on $K$. If $K$ is of characteristic $0$, take $p=1$; otherwise, $p=\operatorname{char} K$. Show that $\mu_\infty$ is non-canonically isomorphic to $\Bbb{Q}/\Bbb{Z}\left[1/p\right]$.

In the case of complex numbers, I see that the first isomorphism theorem given that for $\phi:\Bbb{Q}\to S^1,\phi(\theta)=e^{2\pi i\theta}$, we have $\Bbb{Q}/\operatorname{ker}\phi=\Bbb{Q}/\Bbb{Z}≅\mu_\infty$. Can someone help find a generalization of this homomorphism?

Answer 1

$$\Bbb{Q}/\Bbb{Z}\left[\frac{1}{p}\right] \cong\bigcup_{n\ge 1} \frac1{p^n-1}\Bbb{Z}/\Bbb{Z}\cong \bigcup_{n\ge 1}\Bbb{F}_{p^n}^\times$$